Mathematical reasoning and proof

Mathematics is about problems, and problems must be made the focus of a students mathematical life. Painful and creatively frustrating as it may be, students and their teachers should at all times be engaged in the process-having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other's work."-Lockhart (2002, p. 16) Lockhart (2002, p. 2), a research mathematician, describes the present system of mathematics education at school as a 'nightmare' claiming that it destroys children's 'natural curiosity and love of pattern-making'. He goes further in his critique of school mathematics by saying that 'no actual mathematics' (p. 14) is being done at school, and in the 'place of discovery and exploration' (p. 15) there is only mindless drill and exercise of given rules and algorithms.

In the article, the philosophical significance of quantum computation theory for philosophy of mathematics is discussed. In particular, I examine the notion of "quantum-assisted proof" (QAP); the discussion sheds light on the problem of the nature of mathematical proof; the potential empirical aspects of mathematics and the realism-antirealism debate (in the context of the indispensability argument). I present a quasi-empiricist account of QAP's, and discuss the possible impact on the discussions centered around the Enhanced Indispensabity Argument (EIA).

This is a small (98 page) textbook designed to teach mathematics and computer science students the basics of how to read and construct proofs.

Why do students take the instruction "prove" in examinations to mean "go to the next question"? Because they have not been shown the simple techniques of how to do it. Mathematicians meanwhile generate a mystique of proof, as if it requires an inborn and unteachable genius. True, creating research-level proofs does require talent; but reading and understanding the proof that the square of an even number is even is within the capacity of most mortals.

Proof in Mathematics: an Introduction takes a straightforward, no nonsense approach to explaining the core technique of mathematics.

"Mathematics teaches you to think" is often an empty marketing slogan. With this book, it can become a reality.

This column will publish short (from just a few paragraphs to ten or so pages), lively and intriguing computer-related mathematics vignettes. These vignettes or snapshots should illustrate ways in which computer environments have transformed the practice of mathematics or mathematics pedagogy.

This paper brings up some important points about logic, e.g., mathematical logic, and also an inconsistence in logic as per Gödel's incompleteness theorems which state that there are mathematical truths that are not decidable or provable. These incompleteness theorems have shaken the solid foundation of mathematics where innumerable proofs and theorems have a place of pride. The great mathematician David Hilbert had been much disturbed by them. There are much long unsolved famous conjectures in mathematics, e.g., the twin primes conjecture, the Goldbach conjecture, the Riemann hypothesis, etc. Perhaps, by Gödel's incompleteness theorems the proofs for these famous conjectures will not be possible and the numerous mathematicians attempting to find the solutions for these conjectures are simply banging their heads against the metaphorical wall. Besides mathematics, Gödel's incompleteness theorems will have ramifications in other areas involving logic. This paper looks at the ramifications of the incompleteness theorems, which pose the serious problem of inconsistency, and offers a solution to this dilemma. The paper also looks into the apparent inconsistence of the axiomatic method in mathematics. [Published in international mathematics journal. Acknowledgments: The author expresses his gratitude to the referees and the Editor-in-Chief for their valuable comments in strengthening the contents of this paper.]

We point out seven features that appear to partly constitute a distinctive rhetorical style in which mathematical proofs are written.  As evidence for this, we report on a mathematics journal survey and on interviews with eight mathematicians.  We also report on a survey of U.S. undergraduate "transition course" textbooks that suggest this style is part of the implicit, but not the explicit, curriculum.  We conjecture that this style is useful in minimizing certain kinds of validation errors, i.e., errors in checking the correctness of proofs, and is thus important in learning to construct proofs.  Finally, we suggest that it might be useful to treat this style explicitly when introducing undergraduate students to writing proofs.

Los recursos tecnológicos digitales han mostrado la posibilidad de profundizar en la naturaleza del conocimiento matemático y establecer nexos entre diferentes conceptos. La investigación didáctica ha mostrado que es posible usarlos para apoyar el aprendizaje de la demostración en geometría. La investigación en este campo se ha enfocado ampliamente a la geometría plana, dejando poco explorado el contexto de la geometría espacial. En este taller presentamos un conjunto de problemas de construcción de geometría espacial resueltos con GeoGebra 3d, dirigido a estudiantes con alta capacidad matemática de últimos cursos de Primaria y primeros cursos de Secundaria, con los que se pretende ayudarles en el aprendizaje de la demostración.

First days of a logic course
This short paper sketches one logician’s opinion of some basic ideas that should be presented on the first days of any logic course. It treats the nature and goals of logic. It discusses what a student can hope to achieve through study of logic. And it warns of problems and obstacles a student will have to overcome or learn to live with. It introduces several key terms that a student will encounter in logic.

This article outlines teaching ideas appropriate for primary mathematics. It is mainly aimed at primary school teachers and teacher-researchers. James Russo shares his experiences of exploring proof with a group of 8-and 9-year old students in an Australian primary school.

In this chapter,  the traditional approach of introducing proof in geometry as a means of verification is critiqued from a philosophical as well as a psychological point of view, and in its place an alternative approach to the introduction of proof (in a dynamic geometry environment) is proposed that instead first focuses on the explanatory function of proof.

Mathematicians distinguish between proofs that explain their results and those that merely prove. This paper explores the nature of explanatory proofs, their role in mathematical practice, and some of the reasons why philosophers should care about them. Among the questions addressed are the following: what kinds of proofs are generally explanatory (or not)? What makes a proof explanatory? Do all mathematical explanations involve proof in an essential way? Are there really such things as explanatory proofs, and if so, how do they relate to the sorts of explanation encountered in philosophy of science and metaphysics?

This is the first page of the paper "Coloring the Fourth Dimension. Coloring Polytopes and Complex Curves at the End of the Nineteenth Century", which deals with the epistemological roles and functions of color in mathematics during the 19th century., concentrating on two examples: the well known example of 4-dimensional polytopes, and the less well known example of complex curves.

* Published in: Science in Color: Visualizing Achromatic Knowlegde (ed. by: Bettina Bock von Wulfingen), 2019, de Gruyter, p. 81-97

Tarski’s proof of the law of identity

  Tarski’s LEIBNIZ’S LAW [Introduction to Logic, Sect. 17] is the second-order sentence in variable-enhanced English:

For everything x, for everything y: x = y iff
x has every property y has and y has every property x has.

  The Law of Identity, Tarski’s LAW OF REFLEXIVITY FOR IDENTITY, is:

Everything is equal to itself. For everything x: x = x.

  Tarski says it can be “derived” from Leibniz’s Law. Indeed, the paragraph following its statement begins with the word “PROOF” and ends with the words “which was to be proved”. This paper analyzes Tarski’s argumentation. We found several errors, four noted below.

Mathematicians distinguish between proofs that explain their results and those that merely prove. This paper explores the nature of explanatory proofs, their role in mathematical practice, and some of the reasons why philosophers should care about them. Among the questions addressed are the following: what kinds of proofs are generally explanatory (or not)? What makes a proof explanatory? Do all mathematical explanations involve proof in an essential way? Are there really such things as explanatory proofs, and if so, how do they relate to the sorts of explanation encountered in philosophy of science and metaphysics?

Mathematical induction is a proof technique that can be applied to establish the veracity of mathematical statements. This professional practice paper offers insight into mathematical induction as it pertains to the Australian Curriculum: Mathematics (ACMSM065, ACMSM066) and implications for how secondary teachers might approach this technique to students. In particular, literature on proof – and specifically, mathematical induction – will be presented, and several worked examples will outline the key steps involved in solving problems. After various teaching and learning caveats have been explored, the paper will conclude with some mathematical induction example problems that can be used in the secondary classroom.

Although traditionally neglected, mathematical diagrams have recently begun to attract attention from philosophers of mathematics. By now, the literature includes several case studies investigating the role of diagrams both in discovery and justification. Certain preliminary questions have, however, been mostly bypassed. What are diagrams exactly? Are there different types of diagrams? In the scholarly literature, the term "mathematical diagram" is used in diverse ways. I propose a working definition that carves out the phenomena that are of most importance for a taxonomy of diagrams in the context of a practice-based philosophy of mathematics, privileging examples from contemporary mathematics. In doing so, I move away from vague, ordinary notions. I define mathematical diagrams as forming notational systems and as being geometric/topological representations or two-dimensional representations (or both). I also examine the relationship between mathematical diagrams and spatiotemporal intuition. By proposing an explication of diagrams, I explain (away) certain controversies in the existing literature. Moreover, I shed light on why mathematical diagrams are so effective in certain instances, and, at other times, dangerously misleading.

Examples play a critical role in mathematical practice, particularly in the exploration of conjectures and in the subsequent development of proofs. Although proof has been an object of extensive study, the role that examples play in the process of exploring and proving conjectures has not received the same attention. In this paper, we present a framework that characterizes ways in which mathematicians utilize examples when investigating conjectures and developing proofs. The data consist of 133 mathematicians' responses to two open-ended survey questions. The framework offers categories for the types of examples, uses of examples, and example strategies that mathematicians discussed in reference to their work with conjectures. In addition to presenting the framework, we also discuss potential educational implications of the results.

This paper is concerned with undergraduate and graduate students’ problem solving as they encounter it in attempting to prove theorems, mainly to satisfy their professors in their courses, but also as they conduct original research for theses and dissertations. We take Schoenfeld’s (1985) view of problem, namely, a mathematical task is a problem for an individual if that person does not already know a method of solution for that task. Thus, a given task may be a problem for one individual, who does not already know a solution method for that task, or it may be an exercise for an individual who already knows a procedure or an algorithm for solving that task.

Our study is exploratory. We introduce a possibility on how the processes of experimentation with technology and algebraic approaches may be combined to investigate Ruffini’s rule. We highlight simulations, visual aspects, conjectures, confirmations and enunciations we may develop through the use of the software Winplot, exploring dynamically multiple representation of polynomial functions. We began our approach with a potential conjecture we may have using Winplot when we explore third-degree polynomial functions, leading us to Ruffini’s rule. We enunciate the rule as a preposition: “If the sum of the characters a  0, b, c e d of the polynomial expression p(x) = ax^3 + bx^2 + cx + d is equal to zero, then 1 a root of the polynomial expression”.  Seeking to improve our convincement toward the preposition we enunciated, we explore algebraically and experimentally several notions such as division between polynomials and the theorems of D’Alambert, among others. It allow us to propose in a justified way a possibility to investigate Ruffini’s rule using the Winplot. Basically, we plot a function represented by an algebraic expression formed by the division between a third-degree polynomial and the polynomial (x – 1), that is, f(x) = (ax^3 + bx^2 + cx + d)/ (x – 1). Using the animating tool of Winplot, we may vary or change the real characters in a way that a + b + c + d = 0. On these conditions, the shape of the curve that represents f(x) is a parabola, and it showing the divisibility between the polynomials. Based on our study, that fact convinces us toward the truth of the preposition we enunciated.
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Em Português - Brasil:
Nosso estudo é exploratório. Nós apresentamos uma possibilidade sobre como podem ser engendrados os processos de experimentação com tecnologias e abordagens algébricas na investigação do algoritmo de Briot-Ruffini. Destacamos simulações, aspectos visuais, conjecturas, verificações e enunciações que podem ser realizadas apartir do uso do software Winplot na exploração dinâmica e multi-representacional de funções polinimonais. Iniciamos nossa abordagem com uma potencial conjectura elaborada com o Winplot ao explorarmos funções polinomiais cúbicas, que nos remete ao dispositivo de Briot-Ruffini. Assumimos tal conjectura como uma preposição, a qual enunciamos: “Se a soma dos coeficientes reais a  0, b, c e d de um polinômio p(x) = ax^3 + bx^2 + cx + d for nula, então 1 é raiz do polinômio”. Na busca por aprimorar nosso convencimento acerca da preposição enunciada, exploramos de forma algébrica e/ou experimental diversificadas noções como divisão entre polinôminos e os teoremas de D’Alambert, do Fator e das Raízes Racionais. Isso nos permite propor de forma justificada uma possibilidade para investigarmos o algoritmo de Briot-Ruffini com o Winplot. Basicamente, plotamos uma função representada por uma expressão algébrica composta pela a divisão entre um polinômio do terceiro grau em sua forma genérica e o polinômio (x – 1), ou seja, f(x) = (ax^3 + bx^2 + cx + d)/(x – 1). Ao animarmos com uma ferramenta do Winplot os coeficientes reais de modo que a + b + c + d = 0, a curva que representa o gráfico de f(x) assume a forma de uma parábola, o que mostra a divisibilidade entre os polinômios. Mediante do que fora estudado, esse fato nos convence acerca da verdacidade da preposição enunciada.

Esirgeyen ve Bağışlayan Tanrı'nın adıyla
12/20/1993

Sayın Carl Sagan,

Üç yıl önce, "İlişki" adlı romanınızı okuduğumdan beri, si-zinle "ilişki" kurmayı düşünüyordum. Geçenlerde, dizi halinde yayınlanan yazılarınız beni bu mektubu yazmaya ve İngilizce ilk kitaplarımı size göndermeye zorladı.

Birbirimiz hakkında karşılıklı bilgi edinmemiz için, ken-dimle ilgili başlıca bilgileri özlü cümleler halinde verece-ğim.

Yaşamımın kısa kesitleriyle ilgili, gramer yönünden kısıtlı (İsterseniz "anlatım özürlü") İngilizceyle verdiğim aşağı-daki gelişigüzel bilgiler, dünyaca ünlü bir bilim adamı için birtakım saçma bilgi kırpıntıları olacaktır. Ama, saçmalık-lardan hoşlandığınıza bahse girerim. Bunu yüzünüzdeki muzip gülümsemeden biliyorum.

Yine de,

İngilizce ilk iki kitapçığımı ekte sunuyorum: "Müslüman Din Adamlarına 19 Soru" ve "Hıristiyan Din Adamlarına 19 Soru". Bunları, "Tanrıtanımazlar İçin 19 Soru" ile tamamlamayı tasarlıyorum.

Lütfen, hiç olmazsa son soruyu okuyun (her ikisinde de aşağı yukarı birbirinin benzeridir). Bu soru, Kuran'ın "mucizevi" matematiksel yapısı üzerinedir. Kolayca Hurufilik ile karıştırılmasından dolayı, lütfen bunu incelemeden bir kenara atmayın. Astronominin, Astrolojinin kolu olduğu-nu sanan bir kişiye ne tepki gösterirsiniz?

Edip Yüksel
Arizona Üniversitesi

I am Kishlaya Jaiswal, a high school student (at the time of writing this document), and I have a great curiosity in Research and Mathematics. In this paper, I’ll be presenting and proving the Vandermonde’s Identity. I’ll also be discussing some  other interesting consequences of this identity. And any resemblance of my work somewhere else will be purely coincidental.

Teacher is the pivot of the whole educational process.The present study attempts to compare the emotional intelligence of senior secondary teachers in relation to their teaching aptitude. For this purpose, the investigator adopted random sampling technique to select 300 senior secondary school teachers of Dehradun district. 'Teachers' Emotional Intelligence Inventory' developed by Dr. (Mrs.) Shubhra Mangal and Teaching Aptitude Test' developed by Prof. S.C. Gakhar and Dr. Rajnish was used for data collection. The collected data was analyzed using Mean, S.D. and F-test. The study revealed that all the senior secondary teachers have below average emotional intelligence and average teaching aptitude. Rural and urban senior secondary teachers of high teaching aptitude have shown higher emotional intelligence. Rural senior secondary female teachers have shown better emotional intelligence. Rural senior secondary male and female teachers having high teaching aptitude have shown higher emotional intelligence. Rural senior secondary female teachers of high teaching aptitude have exhibited the highest emotional intelligence. Urban senior secondary male and female teachers having high teaching aptitude have shown higher emotional intelligence. Senior secondary male and female teachers of high teaching aptitude have exhibited better emotional intelligence. Introduction Education plays a very important role in the development of a child's personality. Teacher is the central figure in the educational process which helps in making an individual a better human being. In present times, the plight and role of teacher has changed to a great extent. These changes put an adverse effect on the teaching aptitude of the teachers which in turns affects the holistic development of the students. To overcome these problems, we need such teachers who are capable to balance and control their emotions. This has given rise to the concept of emotional intelligence. Emotional intelligence is that ability which identify; asses and control the emotions of oneself and of others. Goleman, D. (1995) defined emotional intelligence as the capacity to recognize our own feelings and those of others for motivating ourselves and for managing emotions well in ourselves and in our relationships. The emotional intelligence comprises of four mental processes i.e. perceiving and identifying emotions, integrating emotions into thought patterns, understanding one's own and others' emotions and managing emotions (Salovey, P. and Mayer, J., 1990).Emotional intelligence implies a sense of psychological well-being and an ability to skillfully, creatively and confidently adapt in an uncertain, unstructured and changing

This is a text for a course that introduces math majors and math-education majors to the basic concepts, reasoning patterns, and language skills that are fundamental to higher mathematics. The skills include the ability to

    read with comprehension,
    express mathematical thoughts clearly,
    reason logically,
    recognize and employ common patterns of mathematical thought, and
    read and write proofs.  To see another section, see Section 2.2 on "Existence Statements and Negation" on Academia.edu.

The text is amazingly inexpensive. See about that at
http://estymath.com/Proof.html

By tertiary level, in this chapter, we will be referring to undergraduate students majoring in mathematics, including preservice secondary mathematics teachers. Also, in so far as there is information, this chapter will also deal with undergraduate students who major in other subjects and take courses having proofs, such as engineering and computer science majors. It will also consider inservice secondary teachers, as well as other professionals such as engineers, who take additional mathematics courses that deal with proof. In addition, tertiary level will be interpreted to include Masters and Ph.D. students in mathematics and mathematics education.

Para creer un enunciado necesitamos motivos. ¿Está más allá de toda duda, o es razonable creer y simultáneamente dudar de él? Y si podemos creer y dudar de un grupo de enunciados, ¿hay enunciados más creíbles o menos creíbles que otros? ¿Hay grados en las creencias? ¿Son mensurables esos grados y cómo podemos medirlos? Éstas y otras preguntas discutimos en el libro, con un simbolismo y método de análisis cercano al de la lógica, con resultados controlables. Los temas fundamentales son los siguientes: Juegos retóricos en general y sus especies, el discurso oracular, la polémica y el diálogo cooperativo. Condiciones de posibilidad de cada uno de esos juegos retóricos, las condiciones que permiten la existencia del diálogo cooperativo. Las formas de la razón (parte central del libro), el "principio del silogismo", el sistema de la ciencia, sus formas y niveles de fundamentación. La noción de razón suficiente y su generalización en la ciencia popperiana en sentido lato. Los diálogos de fundamentación (lógica dialógica de Lorenzen y Lorenz), exposición crítica de los principios clásicos de la lógica y sus grados de fundamentación y validez. La razón insuficiente y sus diferentes formas. Los límites de la crítica popperiana a la inducción falible. Formas de fundamentación de tesis filosóficas clásicas. Precisiones, comentarios, epílogo.
Versión en línea en www.ciencias.org.ar/cuadernos/CEF.
Noticia en la Académie Internationale de Philosophie des Sciences (AIPS): http://lesacademies.org/fr/a-i-p-s/nos-publications/articles-partages
Résumé: Nous avons besoin de motifs pour croire à un énoncé quelconque. Est'il au-delà de tout doute, ou bien est' il raisonnable de croire et simultanément de mettre en doute sa vérité? Et si nous pouvons croire et douter d'un ensemble d'énoncés, y a-t-il des énoncés plus vraisemblables et moins vraisemblables? Y a-t-il des degrés dans les croyances? Les degrés sont-ils des croyances mesurables et comment pouvons nous les mesurer? Ces questions-ci et d'autres sont traitées dans ce livre, avec un symbolisme et une méthode d'analyse similaire à celle de la logique, avec des résultats contrôlables. Les sujets fondamentaux sont les suivants: les jeux rhétoriques en général et leurs espèces, le discours d'oracle, la polémique et le dialogue coopératif. Les conditions de possibilité de chacun de ces jeux rhétoriques, les conditions qui permettent l'existence du dialogue coopératif. Les genres de la raison (part principale du livre), le «principe du syllogisme», le système de la science, leurs espèces et niveaux de justification. La notion de raison suffisante et sa généralisation au sens large chez Popper. Les dialogues de justification (logique dialogique de Lorenzen et Lorenz), un exposé critique des principes de la logique et ses degrés de justification et validité. La raison insuffisante et leurs espèces. Les limites de la critique Popperienne à l'induction faillible. Les formes de justification de thèses philosophiques classiques. Spécifications, commentaires, épilogue.

This paper aims to clarify Merleau-Ponty's contribution to an embodied-enactive account of mathematical cognition. I first identify the main points of interest in the current discussions of embodied higher cognition and explain how they relate to Merleau-Ponty and his sources, in particular Husserl's late works. Subsequently, I explain these convergences in greater detail by more specifically discussing the domains of geometry and algebra and by clarifying the role of gestalt psychology in Merleau-Ponty's account. Beyond that, I explain how, for Merleau-Ponty, mathematical cognition requires not only the presence and actual manipulation of some concrete perceptible symbols but, more strongly, how it is fundamentally linked to the structural transformation of the concrete configurations of symbolic systems to which these symbols appertain. Furthermore, I fill a gap in the literature by explaining Merleau-Ponty's claim that these structural transformations are operated through motor intentionality. This makes it possible, in turn, to contrast Merleau-Ponty's approach to ontologically idealistic and realistic views on mathematical objects. On Merleau-Ponty's account, mathematical objects are relational entities, that is, gestalts that necessarily imply situated cognizers to whom they afford a specific type of engagement in the world and on whom they depend in their eventual structural transformations. I argue that, by attributing a strongly constitutive role to phenomenal configurations and their motor transformation in mathematical thinking, Merleau-Ponty contributes to clarifying the worldly, historical, and socio-cultural aspects of mathematical truths without compromising what we perceive as their universality, certainty, and necessity.

Abstrak.  Kemampuan siswa dalam pembuktian matematika memegang peranan penting pembelajaran matematika
yang telah menjadi perhatian banyak pihak terutama ahli pendidikan matematika. Oleh karena itu, masalah -masalah
pembuktian  perlu  diberikan  sejak  masih  di  tingkat  dasar  terutama  pada  masalah  geometri  dasar.  Hal  ini  dapat
merangsang penalaran siswa dengan memfasilitasi siswa untuk dalam mengembangkan kemampuan penalaran dan
kemampuan  spasial  dalam  memisahkan,  menggambar,  memodelkan,  menemukan,  mengukur  dan  mengkonstruksi
hubungan-hubungan  geometri.  Salah  satu  materi  geometri  tingkat  dasar  yang  dapat  digunakan  dalam  masalah
pembuktian  adalah  konservasi  luas  yang  merupakan  materi  prasyarat  sebelum  siswa  belajar  tentang  luas  bangun
datar.

Penelitian ini bertujuan untuk  mendeskripsikan strategi pembuktian siswa SD pada masalah konservasi luas
bangun datar ditinjau dari perbedaan gaya kognitif FI dan FD. Peneliti melakukan wawancara terhadap 2 siswa yang
merupakan  subjek  penelitian  kelas  5  SD  mengenai  strategi  pembuktian  konservasi  luas  bangun  datar  dengan
melakukan  tes  GEFT  sebelumnya.  Selanjutnya  data  yang  diperoleh  dianalisis  dengan  menggunakan  triangulasi
waktu.

Hasil dari penelitian ini adalah deskripsi perbandingan strategi siswa SD dalam membuktikan konservasi luas
bangun datar  pada empat tipe masalah pembuktian konservasi luas meliputi: (1)  kedua bangun datar yang diberikan
mempunyai pola dan susunan yang sama, (2)  hanya salah satu bangun datar diketahui polanya, (3)  hanya salah satu
bangun datar yang diketahui polanya tetapi susunannya tidak diketahui secara langsung, dan (4) kedua bangun datar
tidak diketahui susunan dan polanya.

Kata Kunci: Strategi Pembuktian, Konservasi Luas, Gaya Kognitif.

Este trabalho pretende estudar as concepções e práticas de três professores do primeiro ciclo do ensino básico, pertencentes a três gerações diferentes, acerca da resolução de problemas, raciocínio e comunicação. Para isso procura dar resposta às seguintes questões: (1) Como encaram e como valorizam os professores a resolução de problemas, o raciocínio e a comunicação, como concretizam estas orientações curriculares na prática profissional e que dificuldades eventualmente sentem neste domínio? (2) Que factores relativos à sua atitude perante a profissão, à sua relação com a Matemática e à sua visão da aprendizagem os influenciam mais directamente e explicam o seu posicionamento?
Nesta investigação utilizou-se uma metodologia de características qualitativas na variante de estudo de caso em que a recolha de dados se baseou em entrevistas, na observação directa de um conjunto de aulas de cada professor e na análise documental.
A resolução de problemas é entendida pelos três professores como uma actividade que contribui para o desenvolvimento do raciocínio, possibilitando também o tratamento dos diferentes conceitos. No entanto, esta actividade não ocupa o mesmo lugar nas suas práticas. Para uma das professoras a resolução de problemas constitui a actividade fundamental nas suas aulas, existindo a preocupação de atender às vivências dos alunos e aos processos por eles utilizados durante o processo de resolução. Os outros dois professores revelaram possuir um reduzido conhecimento sobre esta actividade, atribuindo pouca importância ao processo de resolução, privilegiando essencialmente o seu papel como forma de aplicar os conceitos previamente ensinados.
Para os três professores a exploração de actividades desafiadoras e não rotineiras é a estratégia adequada para o desenvolvimento da capacidade de raciocínio. Uma professora valoriza os passos seguidos pelos seus alunos, solicitando que estes explicitem os seus raciocínios. Os outros dois professores revelam dificuldades em propor actividades com estas características e em acompanhar os raciocínios utilizados pelos alunos. Em algumas ocasiões, as situações de sala de aula são geridas de modo que os problemas sejam resolvidos de acordo com a estratégia do professor.
No que se refere à comunicação na sala de aula uma das professoras dá grande relevo à discussão, promovendo a negociação como forma de se chegar a consenso. Os dois professores mais jovens, embora considerem que a discussão constitui um excelente meio para ligar a Matemática à Língua Portuguesa e promover o confronto de ideias, revelaram dificuldade em gerir a discussão na sala de aula e em muitos casos limitaram-se a promover um diálogo professor-aluno.
A relação com a Matemática parece influenciar positiva e negativamente a forma como os professores exploram as diferentes situações na sala de aula. Uma das professoras apresenta uma visão da aprendizagem centrada no aluno, organizando as suas práticas em função dos seus interesses e considerando-o o principal dinamizador das actividades. Para os outros dois professores, as actividades dos alunos realizam-se na sequência de propostas do professor a quem cabe o papel central na dinamização da sala de aula. Estes professores manifestam também que a sua principal preocupação se relaciona com o cumprimento do programa.
Quanto à atitude perante a profissão, uma das professoras não mostra um grande envolvimento, apontando como razões a sua curta experiência, a reduzida formação e a falta de colaboração das colegas, preparando as suas aulas a partir de situações que retira dos manuais. Outro professor manifesta alguma desilusão perante a profissão, por considerar que o seu trabalho não é reconhecido pela sociedade, não mostrando grande interesse em introduzir alterações na sua prática lectiva. Finalmente, a professora cujas posições estão mais concordantes com as novas orientações curriculares vive a profissão numa atitude de permanente formação, envolvendo-se em diversas actividades, com reflexos positivos na sua prática pedagógica.

Gauss's quadratic reciprocity theorem is among the most important results in the history of number theory. It's also among the most mysterious: since its discovery in the late 18th century, mathematicians have regarded reciprocity as a deeply surprising fact in need of explanation. Intriguingly, though, there's little agreement on how the theorem is best explained. Two quite different kinds of proof are most often praised as explanatory: an elementary argument that gives the theorem an intuitive geometric interpretation, due to Gauss and Eisenstein, and a sophisticated proof using algebraic number theory, due to Hilbert. Philosophers have yet to look carefully at such explanatory disagreements in mathematics. I do so here. According to the view I defend, there are two important explanatory virtues-depth and transparency-which different proofs (and other potential explanations) possess to different degrees. Although not mutually exclusive in principle, the packages of features associated with the two stand in some tension with one another, so that very deep explanations are rarely transparent , and vice versa. After developing the theory of depth and transparency and applying it to the case of quadratic reciprocity, I draw some morals about the nature of mathematical explanation.

Öz Bu çalışma 7. sınıf ortaokul öğrencilerinin Matematik dersindeki muhakeme etme beceri düzeylerinin belirlenmesi amacıyla yapılmıştır. Çalışma, 2015-2016 eğitim öğretim yılının birinci döneminde, Türkiye'nin Karadeniz bölgesinde bulunan bir devlet okulunda öğrenim gören toplam 94 öğrenci ile gerçekleştirilmiştir. Çalışmada veri toplama aracı olarak Erdem'in (2015) Zenginleştirilmiş Öğrenme Ortamının Matematiksel Muhakemeye ve Tutuma Etkisi adlı doktora tezinde geliştirmiş olduğu iki problem durumu kullanılmıştır. Problem TIMSS'in (2003) matematiksel muhakeme aşamalarına göre analiz edilmiştir. Veriler içerik analizi ile değerlendirilmiştir. Sonuç olarak öğrencilerin problemleri çözerken istenilen ve yazılanları belirterek muhakeme becerilerini ortaya koydukları belirlenmiştir. Bazı öğrencilerin ise tablo yöntemini kullandıkları belirlenmiştir. Bir kısım öğrencinin ise soruyu anlayamadığı ve doğru cevaba ulaşamadığı tespit edilmiştir. Anahtar Kelimeler: TIMSS, Muhakeme Etme, Ortaokul, Matematik Dersi, Matematiksel Muhakeme. Abstract This study was conducted with the aim of determining the reasoning skill levels of 7th grade secondary school students in mathematics course. The study was carried out with 94 students in total, who study in a state school in the Black Sea region of Turkey during the first semester of the academic year of 2015-2016. Two problematic situations developed by Erdem (2015) in his doctoral thesis, which is named as the Effect of Enriched Learning Environment on Mathematical Reasoning and Attitude, have been used in this study. Problematic situation were analyzed according to the mathematical reasoning phases of TIMSS (2003). The data retrieved in the study were evaluated via content analysis. Consequently, examining the answers given by students to problems, it is determined that the students presented their reasoning skills through indicating what was asked and given in the question while solving the problems. This indicates that the students used spatial reasoning. It was also determined that some students used the table method. A part of the students did not understand the questions, thus failed to obtain the correct answer.

Ethics and mathematics have long invited comparisons. On the one hand, both ethical and mathematical propositions can appear to be knowable a priori, if knowable at all. On the other hand, mathematical propositions seem to admit of proof, and to enter into empirical scientific theories, in a way that ethical propositions do not. In this article, I discuss apparent similarities and differences between ethical (i.e., moral) and mathematical knowledge, realistically construed -- i.e., construed as independent of human mind and languages. I argue that some are are merely apparent, while others are of little consequence. There is a difference between the cases. But it is not an epistemological difference per se. The difference, surprisingly, is that ethical knowledge, if it is practical, cannot fail to be objective in a way that factual knowledge can. One upshot of the discussion is radicalization of Moore’s Open Question Argument. Another is that the concepts of realism and objectivity, which are widely identified, are actually in tension.

On pense généralement que l'impossibilité, l'incomplétdulité, la paracohérence, l'indécidabilité, le hasard, la calcul, le paradoxe, l'incertitude et les limites de la raison sont des questions scientifiques physiques ou mathématiques disparates ayant peu ou rien dans terrain d'entente. Je suggère qu'ils sont en grande partie des problèmes philosophiques standard (c.-à-d., jeux de langue) qui ont été la plupart du temps résolus par Wittgenstein plus de 80 ans.
Je fournis un bref résumé de quelques-unes des principales conclusions de deux des plus éminents étudiants du comportement des temps modernes, Ludwig Wittgenstein et John Searle, sur la structure logique de l'intentionnalité (esprit, langue, comportement), en prenant comme point de départ La découverte fondamentale de Wittgenstein, à savoir que tous les problèmes véritablement « philosophiques » sont les mêmes, les confusions sur la façon d'utiliser la langue dans un contexte particulier, et donc toutes les solutions sont les mêmes— en regardant comment la langue peut être utilisée dans le contexte en cause afin que sa vérité (Conditions de satisfaction ou COS) sont claires. Le problème fondamental est que l'on peut dire n'importe quoi, mais on ne peut pas signifier (état clair COS pour) toute déclaration arbitraire et le sens n'est possible que dans un contexte très spécifique.
Je dissé que quelques écrits de quelques-uns des principaux commentateurs sur ces questions d'un point de vue wittgensteinien dans le cadre de la perspective moderne des deux systèmes de pensée (popularisé comme «penser vite, penser lentement»), en utilisant une nouvelle table de intentionnalité et la nomenclature de nouveaux systèmes doubles. Je montre qu'il s'agit d'un puissant heuristique pour décrire la vraie nature de ces questions scientifiques, physiques ou mathématiques putatives qui sont vraiment mieux abordés comme des problèmes philosophiques standard de la façon dont la langue doit être utilisée (jeux de langue dans Wittgenstein terminologie).

This paper discusses the nature of mathematical induction, and analyses it mathematically and psychologically, including behavioural and conceptual dimensions and prerequisites, and  common errors and misconceptions.  It analyses the treatment of MI in a selection of school and college texts (the selection is rather dated now). Overall a multifaceted pedagogical discussion of MI.

This paper shows why the non-trivial zeros of the Riemann zeta function ζ will always be on the critical line Re(s) = 1/2 and not anywhere else on the critical strip bounded by Re(s) = 0 and Re(s) = 1, thus affirming the validity of the Riemann hypothesis. [The paper is published in a journal of number theory.]