Mathematical reasoning and proof

Mathematical inquiry challenges students to ask questions, create definitions and think very carefully about how they are going to solve a problem.

With the emphasis on mathematical reasoning, judgement and problem-solving skills, the Thinking through Mathematics series requires students to investigate questions that are open-ended and ambiguous, rather than closed and defined. The process of reasoning is reinforced as students consider the parameters of their inquiry, formulate, trial and enact a plan, and share their thinking process alongside the results.Each book provides ten mathematical inquiry units, each posing a real-life, ambiguous question for inquiry.

Thinking through Mathematics provides a practical entry into inquiry-based mathematics learning which immerses students in solving authentic, complex problems.

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.

Aristotle's Demonstrative Logic. History and Philosophy of Logic.  30 (2009) 1–20. 

Demonstrative logic is the study of demonstration as opposed to persuasion. It is the subject of Aristotle’s two-volume Analytics, as he said in the first sentence. Many of his examples are geometrical. Every demonstration produces (or confirms) knowledge of (the truth of) its conclusion for every person who comprehends the demonstration. Persuasion merely produces opinion. Aristotle presented a general truth-and-consequence conception of demonstration meant to apply to all demonstrations. According to him, a demonstration is an extended argumentation that begins with premises known to be truths and that involves a chain of reasoning showing by deductively evident steps that its conclusion is a consequence of its premises. In short, a demonstration is a deduction whose premises are known to be true. Aristotle’s general theory of demonstration required a prior general theory of deduction presented in the Prior Analytics. His general immediate-deduction-chaining conception of deduction was meant to apply to all deductions. According to him, any deduction that is not immediately evident is an extended argumentation that involves a chaining of immediately evident steps that shows its final conclusion to follow logically from its premises.

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.

Symbols represent by codes like conventions, whereas icons represent by similarity (Couturat 1901; Dascal 1978; Gensini 1991; Serfati 2001). Until recently, much of the literature in philosophy of notation tended to follow Leibniz in assuming that we always or most often think in symbols. However, current debates have increasingly been driven by the recognition that any system of representation depends, to some extent, on iconicity, such that the purported line of demarcation is rather a continuum (Stjernfelt 2014). This slow but resolute turn toward iconic signs has begun to change how we see logic. With different emphases, linguists (Simone 1995; Van Langendonck 2007), logicians (Burch 1991; Shin 1994, 2002; Hammer 1995; Allein & Barwise 1996; Pietarinen 2006, 2010, 2011, 2012), historians of mathematics (Nets 2004; Mancosu, Jorgensen & Pedersen 2005; Giaquinto 2007), semioticians (Stjernfelt 2007, 2014; Bordron 2011; Dondero & Fontanille 2012), philosophers of mind (Champagne 2014) and cognitive scientists (Glasgow, Nara y Anan & Chandrasekaran 1995; Hoffmann 2010a, 2010b, 2011; Magnani 2011; Nakatsu 2010) have all recognized that a better understanding of reasoning by icons, specifically diagrams, is crucial to understanding problem-solving, inference-drawing, and hypothesis-making. Arguably, no one has explored the potential of iconic notations more systematically than C. S. Peirce. It is commonly believed that the birth of new formal logic(s) by Frege (1884), Russell (1903) and Couturat (1904) rendered Kant’s appeal to intuition redundant (cf. Kneale & Kneale 1962, VII-VIII; Coffa 1991; Carson & Huber 2006). However, an alternate line of development is clearly discernable in the work of Peirce. For Peirce, the notion of intuition is by no means redundant, being instead the faculty which allows necessary reasoning to yield informative truths (Peirce 1931-1958; Peirce 2010; cf. Hintikka 1980; Hookway 1985; Ketner 1985; Shin 1997; Pietarinen 2006; Stjernfelt 2007; Bellucci 2012). The diagram, in this Peircean paradigm, transforms intuition into a visual commodity amenable to careful public scrutiny. Indeed, one of the most striking features of Peirce’s diagrammatic notation is its depiction of inferences as transformations. Inference rules, being answerable to the self-same nature of images, are motivated in a way that makes them less rule-like. For example, enclosures, a common device used by Peirce, place distinct limits on what counts as included/inside, excluded/outside, or both. One can attempt to transgress these limits, but the iconic sign-vehicles at hand simply repel erroneous interpretations. While Venn exploited this to prove categorical syllogisms, Peirce shows how to generalize the method, thereby giving a novel justification for the normative force of logic. Peirce’s unpublished technical work in logic has thus far influenced debates mainly through the intermediary of specialists (Shin 2002; Sowa 2011), but the upcoming publication of Peirce’s full Existential Graphs (Logic of the Future, edited by A.-V. Pietarinen) promises to augment this rate of influence, by making available a 1000-page buffet of diagrammatic and iconic logical systems. Participants to this symposium are thus invited to reflect on how Peirce-inspired diagrammatic approaches to notation can positively reshape issues pertaining to logic, cognition, and reason-giving practices generally.

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.

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.

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.

Mathematical reasoning has been emphasised as one of the key proficiencies for mathematics in the Australian curriculum since 2011 and in the Canadian curriculum since 2007. This study explores primary teachers’ perceptions of mathematical reasoning at a time of further curriculum change.Twenty-four primary teachers from Canada and
Australia were interviewed after engagement in the first stage of the Mathematical Reasoning Professional Learning Program incorporating demonstration lessons focused on reasoning conducted in their schools. Phenomenographic analysis of interview transcripts
exploring variation in the perceptions of mathematical reasoning held by these teachers revealed seven categories of description based on four dimensions of variation.The categories delineate the different perceptions of mathematical reasoning expressed by the participants of this study. The resulting outcome space establishes a framework that
facilitates tracking of growth in primary teachers’ awareness of aspects of mathematical reasoning

Mathematical reasoning is now featured in the mathematics curriculum documents of many nations, but this necessitates changes to teaching practice and hence a need for professional learning. The development of children's mathematical reasoning requires appropriate encouragement and feedback from their teacher who can only do this if they recognise mathematical reasoning in children's actions and words. As part of a larger study, we explored whether observation of educators conducting mathematics lessons can develop teachers' sensitivity in noticing children's reasoning and consideration of how to support reasoning. In the Mathematical Reasoning Professional Learning Research Program, demonstration lessons were conducted in Australian and Canadian primary classrooms. Data sources included post-lesson group discussions. Observation of demonstration lessons and engagement in post-lesson discussions proved to be effective vehicles for developing a professional eye for noticing children's individual and whole-class reasoning. In particular, the teachers noticed that children struggled to employ mathematical language to communicate their reasoning and viewed limitations in language as a major barrier to increasing the use of mathematical reasoning in their classrooms. Given the focus of teachers' noticing of the limitations in some types of mathematical language, it seems that targeted support is required for teachers to facilitate classroom discourse for reasoning.

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.

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 inquires the ways in which paper folding constitutes a mathematical practice and may prompt a mathematical culture. To do this, we first present and investigate the common mathematical activities shared by this culture, i.e. we present mathematical paper folding as a material reasoning practice. We show that the patterns of mathematical activity observed in mathematical paper folding are, at least since the end of the 19 th century, sufficiently stable to be considered as a practice. Moreover, we will argue that this practice is material. The permitted inferential actions when reasoning by folding are controlled by the physical realities of paper-like material, whilst claims to generality of some reasoning operations are supported by arguments from other mathematical idioms. The controlling structure provided by this material side of the practice is tight enough to allow for non-textual shared standards of argument and wide enough to provide sufficiently many problems for a practice to form. The upshot is that mathematical paper folding is a non-propositional and non-diagrammatic reasoning practice that adds to our understanding of the multi-faceted nature of the epistemic force of mathematical proof. We then draw on what we have learned from our contemplations about paper folding to highlight some lessons about what a study of mathematical cultures entails.

- To be published in Synthese (2019)

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.

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.

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

Discourse relations link two different discourse units into a compound unit, and it is the presence of such relations that gives a discourse coherence. For example, the relations RESULT and NARRATION are responsible for the perceived narrative progression of the events described in the discourse. Building on Hobbs (1985) and Kehler (2002), we use the definitions of RESULT and NARRATION to derive constraints that pertain to the internal structure of discourse units, and we argue that the plausibility of these constraints lends a new type of support for the
definitions that we propose.

Parte delle riflessioni di Wittgenstein sul concetto di prova 1 sono una critica di alcune idee di Frege; quando Frege sostiene che usare le leggi logiche e dubitarne è un tipo di follia, Wittgenstein si domanda che tipo di follia potrebbe essere questa 2 . Molte delle sue osservazioni su giochi linguistici "devianti" o "alieni" sono un tentativo di rispondere a questa domanda 3 . La distanza tra i due autori è dunque profonda, e riguarda principalmente l'atteggiamento verso il platonismo matematico; però Wittgenstein ha sempre mantenuto alcuni principi fregeani, specialmente l'idea di "prova" matematica o logica come qualcosa di normativo, cioè essenzialmente differente da un esperimento (seguito in questo da autori come Kreisel in antagonismo con il "quasi empirismo" matematico di Lakatos). In questo mio saggio vorrei illustrare le idee base di Frege sul concetto di "prova" per mostrare quanto di queste idee è rimasto e quanto è stato abbandonato da Wittgenstein, per concludere con una breve analisi del punto di vista di Wittgenstein sulla prova di Gödel. * Ringrazio Carlo Dalla Pozza, Daniele Chiffi, Morgana Chiappalone, Marcello Frixione e un referee anonimo: le loro letture attente di una versione precedente del lavoro hanno aiutato a renderlo meno confuso e più leggibile. Ogni errore comunque va attribuito all'autore che non sempre ha saputo utilizzare al meglio i consigli e i dubbi postigli. 1 In quanto segue uso "prova" e "dimostrazione" senza particolare distinzione; la maggior parte del lavoro è dedicato quasi esclusivamente al concetto di prova in logica o in matematica. 2 OFM I:151/I:152. Wittgenstein si riferisce a un'osservazione fatta da Frege nell'introduzione ai Principi fondamentali dell'aritmetica del 1893. 3 Su questo problema ho iniziato a lavorare a partire da Penco 2008, che individua in questa reazione a Frege una delle fonti del cosiddetto "antropologismo" di Wittgenstein.

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

Starting with an elementary result for the median triangle of a triangle in plane geometry, this result is generalized by asking 'what-if' questions. Ith therefor gives an example of problem posing, experimentation, conjecturing and proof. The theorems of Ceva, Desargues and Pascal are used to prove the observed conjectures.

CORCORAN ON MATHEMATICAL OPINION AND KNOWLEDGE—draft 13 of a review for MATHEMATICAL REVIEWS of: Paseau, Alexander, Knowledge of mathematics without proof, British J. Philos. Sci. 66 (2015), no. 4, 775--799]
The article under review, “Knowledge of mathematics without proof”—hereafter KMWP—appears in a respected peer-reviewed journal. Its author, an Associate Professor at Oxford University, presented previous versions at conferences in Bergen, Cambridge, Dubrovnik, and Oxford. Its challenging subject, the nature of mathematical knowledge, has been studied and debated since ancient times.

A number of points are brought up below for pondering and possible discussion. We should not regard our physics as the one and only reality. What is described below may seem like science fiction, but should not be brushed off as inconsequential.

Just like any other cultural group, mathematicians like to tell stories. We tell heroic stories about famous mathematicians, to inspire or reinforce our cultural values, and we encase our results in narratives to explain how they are interesting and how they relate to other results. We also tell stories to convince others that our results are valid, and preferably also to explain why they are true. These stories are what you know as "proofs".

During the 2012 Visiting Professorship of Prof. Sundholm in Lille, the logic group of Lille started probing possible ways of implementing Per Martin-Löf's Constructive Type Theory (CTT) in the dialogical perspective. The first publication in particular on the subject—Aarne Ranta's (1988) paper—was read and discussed during Sundholm’s seminar. These discussions strongly suggested that the game-theoretical conception of quantifiers, which marshalls interdependent moves, provides a natural link between CTT and dialogical logic. This idea triggered several publications by the group of Lille in collaboration with Nicolas Clerbout and Juan Redmond at the University of Valparaíso, including that of the (2015) book by Clerbout & Rahman providing a systematic development of this way of linking CTT and the dialogical conception of logic.

However the (Clerbout & Rahman, 2015) book was written from the CTT perspective on dialogical logic, rather than the other way round. The present book, “Immanent Reasoning or Equality in Action”, should provide the other perspective in the dialogue between the dialogical framework and Constructive Type Theory.

In order to develop a dialogical perspective on the links between CTT and dialogical logic we will follow three complementary paths:
A. One of chief ideas animating our study is that Sundholm's (1997)1 notion of epistemic assumption is closely linked to the Copy-Cat Rule or Socratic Rule that distinguishes the dialogical framework from any other game-theoretical approaches. The linked is established via the dialogical understanding of definitional equality.
B. We will join—with some nuances linked to point C below—Martin-Löf’s (2017a,b) suggestions that the new insights provided by the dialogical framework mainly amount to the following three interconnected points:
B.1. the introduction of rules of interaction rather than of inference rules;
B.2. the challenge to what Kuno Lorenz (2014 [1987], p. 71) calls the semantization of pragmatics: deontic features are formalized with the help of specific propositional operators (and indexes) upon which the truth-value of the resulting proposition is made dependent;
B.3. the central role of the notion of execution in the rules of interaction: executions are responses to questions of knowing how.
C. As indicated by the subtitle, “A Plaidoyer for the Play Level”, we will stress the importance of the play level over the strategy level: this binds the point of execution with that of equality.

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.]

This article provides an illustration of the explanatory and discovery functions of proof with an original geometric conjecture made by Clough, a Grade 11 student in relation to Viviani's theorem. After logically explaining (proving) the result geometrically and algebraically, the result is generalised to other polygons by further reflection on the proof(s). Different proofs are given, each giving different insights that lead to further generalisations. The underlying heuristic reasoning is carefully described in order to provide an exemplar for designing learning trajectories to engage students with these functions of proof. Some interactive Java sketches regarding the variation and generalizations are available at: http://frink.machighway.com/~dynamicm/clough.html

Mathematical argumentation and proof (MA&P) traditionally are major topics of mathematics education in secondary and tertiary education. Although many studies focus on MA&P it remains unclear how they contribute to a coherent understanding of MA&P processes. We have analyzed PME research reports focusing on MA&P published 2010 to 2014 to determine the different prerequisites as well as goals of argumentation and proving processes investigated within these reports. Results indicate that research on MA&P covers a broad range of processes, sub-skills and knowledge facets, but that individual reports predominantly address only singular aspects. A holistic approach to MA&P, taking into account the whole process or multiple sub-skills, is rare. We discuss implications for future research of MA&P.

Mathematics can help investigate hidden patterns and structures in music and visual arts. Also, math in and of itself possesses an intrinsic beauty. We can explore such a specific beauty through the comparison of objects and processes in math with objects and processes in the arts. Recent experimental studies investigate the aesthetics of mathematical proofs compared to those of music. We can contextualize these studies within the framework of category theory applied to the arts (cARTegory theory), thanks to the helpfulness of categories for the analysis of transformations and transformations of transformations. This approach can be effective for the pedagogy of mathematics, mathematical music theory, and STEAM.

This study aimed to analyze the difficulties students in constructing
mathematical proof on discrete mathematics course. This study was
conducted in mathematics education student who contracted course
discrete mathematics. Data were collected by test results and interviews
with students. The result of the analyses revealed four main difficulties
faced by students: (1) understanding of mathematical concepts, (2)
language and mathematical notation, (3) strategies of proof, and (4) read of proof. In addition, student perception about mathematics and
mathematical proof construction affected student proof. Writing about a
good proof was another difficulty faced by students.

Aristotle’s Prototype Rule-based Underlying Logic [PRINTER PROOF VERSION]

This expository paper on Aristotle’s prototype underlying logic is intended for a broad audience that includes non-specialists. It requires as background as discussion of Aristotle’s demonstrative logic. As explained in Corcoran (2009), demonstrative logic or apodictics is the study of demonstration as opposed to persuasion. It is the subject of Aristotle’s two-volume Analytics, as its first sentence says. Many of his examples are geometrical. A typical geometrical demonstration requires a theorem that is to be demonstrated, known premises from which the theorem is to be deduced, and a deductive logic by which the steps of the deduction proceed. Every demonstration produces (or confirms) knowledge of (the truth of) its conclusion for every person who comprehends the demonstration. Persuasion merely produces opinion. Aristotle presented a general truth-and-consequence theory of demonstration meant to apply to all demonstrations: a demonstration is an extended argumentation that begins with premises known to be truths and that involves a chain of reasoning showing by deductively evident steps that its conclusion is a consequence of its premises. In short, a demonstration is a deduction whose premises are known to be true. Aristotle’s general theory of demonstration required a prior general theory of deduction presented in the Prior Analytics. His general immediate-deduction-chaining theory of deduction was meant to apply to all deductions: any deduction that is not immediately evident is an extended argumentation that involves a chaining of immediately evident steps that shows its final conclusion to follow logically from its premises. His deductions, both direct and indirect, were rule-based and not tautology-based. The idea of tautology-based deduction, which dominated modern logic in the early years of the 1900s, is nowhere to be found in Analytics. Rule-based deduction was rediscovered by modern logic. To illustrate his general theory of deduction, Aristotle presented a prototype: an ingeniously simple and mathematically precise special case traditionally known as the categorical syllogistic. With reference only to propositions of the four so-called categorical forms, he painstakingly worked out exactly what those immediately evident deductive steps are and how they are chained. In his specialized theory, Aristotle explained how to deduce from a given categorical premise set, no matter how large, any categorical conclusion implied by the given set. He did not extend this treatment to non-categorical deductions, thus setting a program for future logicians. The prototype, categorical syllogistic, was seen by Boole as a “first approximation” to a comprehensive logic. Today, however it is regarded as the first of the dozens of logics already created and as the first exemplification of a genus that continues to expand. 


KEYWORDS: demonstration, deduction, direct, indirect, categorical syllogistic, premise, conclusion, implicant, consequence, belief, knowledge, opinion, cognition, intuitive, demonstrative.
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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.

In pure mathematics, unproved conjectures differ in their strength. Some are well-supported by evidence and regarded as "almost proved", others are weak. Since there is nothing to the relation of pure mathematical propositions but logic, this supports the theory of logical probability (a.k.a. non-deductive logic) - the view of Keynes and Jaynes that (one fundamental kind of) probability is a sort of pure logic; a degree of partial implication. Examples from mathematics are given and conclusions drawn about the nature of logical probability.

Despite the recognized importance of mathematical proof in secondary education, there is a limited but growing body of literature indicating how preservice secondary mathematics teachers (PSMTs) view proof and the teaching of proof. The purpose of this survey research was to investigate how PSMTs in Australia and the United States perceive of proof in the context of secondary mathematics teaching and learning. PSMTs were able to outline various mathematical and pedagogical aspects of proof, including: purposes, characteristics, reasons for teaching, and imposed constraints. In addition, PSMTs attended to differing, though overlapping, features of proof when asked to determine the extent to which proposed arguments constituted proofs or to decide which arguments they might present to students.

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.

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

A proof by mathematical induction demonstrates that a general theorem is necessarily true for all natural numbers. It has been suggested that some theorems may also be proven by a 'visual proof by induction' (Brown, 2010), despite the fact that the image only displays particular cases of the general theorem. In this study we examine the nature of the conclusions drawn from a visual proof by induction. We find that, while most university-educated viewers demonstrate a willingness to generalize the statement to nearby cases not depicted in the image, only viewers who have been trained in formal proof strategies show significantly higher resistance to the suggestion of large-magnitude counterexamples to the theorem. We conclude that for most university-educated adults without proof-training the image serves as the basis of a standard inductive generalization and does not provide the degree of certainty required for mathematical proof.

This study explores the metacognitive skills and mathematical reasoning in mathematics during problem solving tasks. The study uses the analytical framework of Schoenfled (1985) to study the problem solving and reasoning behaviour at the executive or control level. The study is conducted in three phases to explore the objectives in students studying in grade 5 of a Delhi-based school.
In Phase 1, the researcher administers a 12 item Test 1 (N = 40) which caters to four of the basic mathematical domains - measurement, numerical operations, patterns and data representation. The purpose of this phase is to select a sample (N = 12) of high achieving students and to group them into three groups such that the members of each group has an equal mean score of Test 1.
In Phase 2 the researcher designs and administered a 12 days interventions (on the basis of IMPROVE by Mevarech & Kramarski) in which the researcher develops Metacognitive skills while problem solving and reasoning (of 4 skills). At the end of each skill, the skill is tested and it is recorded through field observations to analyse the mathematical reasoning behaviour of the group.
In Phase 3, the researcher conducts a Test 2 (N = 12) similar to Test1 in terms of content, instructions and items. The purpose of the phase is to see the effectiveness of intervention and various other factors.
The intervention helped in exploring the different kinds of metacognitive skills in cooperative settings during problem solving and mathematical reasoning. The results suggest that a continuous interplay of metacognitive and problem solving behaviour appears to be necessary for successful development of mathematical reasoning and maximum student involvement.

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.

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).

Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The  Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case.

Many mathematics departments have instituted transition-to-proof courses for second semester sophomores to help them learn how to construct proofs and to prepare them for proof-based courses, such as abstract algebra and real analysis. We have developed a way of getting students, who often stare at a blank piece of paper not knowing what to do, started on writing proofs. This is the technique of writing proof frameworks, based on the logical structure of the statements of the theorems and associated definitions. Also, in order to unpack the conclusion and know what is to be proved, students need definitions to become " operable " .

Two students from a transition-to-proof course were interviewed as they attempted proofs. One student, Brad, took a referential approach, meaning he used examples to make sense of the concepts. The other student, Carla, took a syntactic approach, meaning she used definitions and logic, but did not use examples. Both made progress on the proofs, but exhibited different strengths and different difficulties. The authors conclude, "Both students seem to have an underdeveloped notion of how to use examples and syntax together to construct a proof." Antonini, S. (2001). Negation in mathematics: Obstacles emerging from an exploratory study. In M. van den Heuvel-Panhuizen (Ed.),