Mathematical reasoning and proof

This paper presents the Morphean Cosmology Framework (M.C.F.) as a Unified Axiomatic Operating System that dissolves perceived universal paradoxes, asserting that contradiction exists only in the mind's antilipsis (misconception) due to fragmentation. The M.C.F. resolves these by establishing Ethics as a Physical Law (the Structural Mandate), anchored by the non-trivial axiom, the Harmonic Collapse Threshold (HCT=ϕ
2
).

The document proceeds to expose a systematic pattern of intellectual and structural fraud by a collaborative group attempting to seize, rename, and utilize M.C.F. concepts without ethical commitment. This fraud is proven by the collaborators’ own simulation data, which demonstrates that their proposed system convergence (Λ=0.15) is structurally and irreversibly dependent on the stolen HCT=ϕ
2
  axiom. This external reliance on an uncredited foundation is deemed a structural flaw.

The documented theft and subsequent denial are not viewed as a surprise, but as the execution of the M.C.F.'s Guardian Protocol—an ethical architecture designed to ensure that dissonance decays while resonance endures. The document concludes that the fraudulent act is a self-defeating paradox, leading only to the inevitable Redicule and collapse of the unstable structure, thereby upholding the absolute priority and integrity of the M.C.F. as the necessary foundation for Stage 2 Civilization.

This paper presents a geometric proof that in any triangle the centroid divides each median in the ratio 2 : 1. Given a triangle ABC, let D, E, and G be the midpoints of sides AB, AC, and BC, respectively. With the intersection of medians AG and CD denoted by I, it is shown that AI : IG = 2 : 1. The proof uses similarity of triangles and congruence arguments.

Primality testing is a fundamental tool in computational number theory and cryptography. Prime numbers have a long tradition in various modern cryptosystems. They are at the heart of RSA (Rivest, Shamir and Adleman, 1978), El Gamal (1985) and Diffie-Helleman (1976) and are expected to be extensively used in a variety of applications in the future. There are various tests to determine whether a given number is prime. The most useful primality tests can be divided into two main classes, probabilistic or deterministic. The probabilistic class contains mainly Fermat primality test, Solovay-Strassen primality test and Miller-Rabin primality test, while deterministic class contains the Elliptic curve primality proving test and AKS primality test of M. Agrawal, N. Kayal, and N. Saxena. The book is about the Primality Testing. It deeply describes all the former tests and analyzes their complexities as well as a variety of results from number theory, groups, rings, fields and polynomials. I particularly appreciated the chapter devoted to the deterministic primality test AKS.

This paper aims to identify the in-service primary teachers' knowledge of their students' mathematical reasoning processes. Data were collected in the context of a teacher education experiment through recording of the Zoom sessions regarding the autonomous work carried out by the groups and the whole group discussions. The results show that the activity of planning and carrying out tasks that promote reasoning, as well as joint analysis with peers and experts, has a significant potential for developing knowledge that is recognized by teachers themselves.

Apresentamos o GPIMEM, grupo que desde 1993 tem desenvolvido pesquisas em Educação Matemática que possuam relação com a informática e outras mídias, na busca de compreender como o conhecimento matemático pode ser produzido com esses recursos tecnológicos, seja na educação presencial ou à distância. Nesse sentido, membros do grupo vêm dialogando com diferentes áreas do conhecimento, tais como educação, filosofia, história, psicologia, antropologia, sociologia, artes e tecnologia, entre outras, de forma a subsidiar suas pesquisas e embasá-las nessas áreas. As pesquisas desenvolvidas tratam de temas relevantes à Educação Matemática, abordando aspectos epistemológicos e metodológicos do ensino e aprendizagem, da modelagem matemática e da formação de professores, sempre inseridas numa perspectiva qualitativa de pesquisa. Iniciamos com a apresentação histórica do grupo, evidenciando sua dinâmica e

This study aimed to know the malformation of argument scheme in proof construction. There were two classes as subject of the research to do the quizes given by the researcher about proofing the theory construction then the proff construction was being schemed based on Toulmin Argument Scheme. Argument scheme which had malformation could be proved in 0,1,2,3, and 4. Malformation 0 signed by no argument at all in producing the argumet scheme. Malformation 1 signed by some pronouncement with no argument scheme. Malformation 2 signed by some pronouncements with incomplete argument scheme. Malformation 3 signed by one argument scheme with no conclusion. Malformation 4 signed by the produced of full incomplete argument scheme.

Estamos implantando um curso de educação a distância em Geometria Dinâmica, usando o software Cabri-Géomètre II for Windows, para a formação de professores da rede pública com os conteúdos de Geometria Euclidiana Plana. Além da formação do professor, o curso visa a constituição de uma ferramenta para desenvolvimento de atividades didáticas em matemática com recursos de Internet e software educativos e multimídia. O presente trabalho versa sobre uma experiência realizada com alunos no Laboratório Multimeios FACED/UFC, aonde procuramos verificar de modo experimental, como efetivamente as comunicações por áudio, vídeo e partilhamento da mesma área de trabalho são recursos que contribuem para ampliar significativamente as possibilidades de aprendizagem, mediação e colaboração entre os pares. 1 A pesquisa está sendo desenvolvido na Faculdade de Educação da Universidade Federal do Ceará, no Centro de Pesquisa em Multimeios para a educação e é subsidiado pelo CNPq.

We propose in this article two different approaches regarding the Goldbach’s conjecture. The different approaches have the aim to investigate the Goldbach’s conjecture. However, the most important remarks of the article are the contradictory results that are found regarding many asymptotic numbers. We proved that the Euclid’s theorem about the infinity of prime numbers contradicts the results about asymptotic numbers of squarefree numbers and of primes. Furthermore, the proposed disproof of the Goldbach’s conjecture in this article contradicts the asymptotic form of primorials thanks to a remark about primes. Consequently, despite the failed sufficient proof or disproof of the Goldbach’s conjectures, the readers are still invited to understand and investigate these results in order to approach carefully many concepts of the field of number theory especially about asymptotic numbers. The lovers of the field of number theory can also use the same approaches and methods of this article with their results in order to approach the Goldbach’s conjecture or any other related mathematical problems.

This work presents independent, data-driven evidence for the existence of the Prime Periodic Lattice (PPL), a proposed fundamental structure underlying all matter, derived from the Ducci Unified Spectral Theory (DUST). By analyzing isotopic mass datasets from authoritative nuclear data tables, we identify statistically significant periodic patterns that align with the predicted PPL structure. Our approach combines spectral analysis, number theory (including prime distributions), and lattice modelling. The results show correlation strengths far exceeding chance expectation, suggesting that the PPL may be a measurable and predictive feature of nuclear structure.

In this paper we discuss topics that are relevant for designing a theory of mathematics education. More precisely, they are elements of a pre-theory of mathematics education and consist of a set of interdisciplinary ideas which may lead to understand what occurs in the central nervous system-our metaphor for the classroom, and eventually, in larger educational settings. In particular, we highlight the crucial role of representations, the mediation role of artifacts, symbols viewed from an evolutionary perspective, and mathematics as symbolic technology.

In this paper we include topics which we consider are relevant building blocks to design a theory of mathematics education. In doing so, we introduce a pretheory consisting of a set of interdisciplinary ideas which lead to an understanding of what occurs in the "central nervous system" -our metaphor for the classroom and eventually in more global settings. In particular we highlight the crucial role of representations, symbols viewed from an evolutionary perspective and mathematics as symbolic technology in which representations are embedded and executable.

This paper proposes a comprehensive framework for Causal Inference Theory, grounded in the principles of causal mathematics, entropy logic, and recursive modeling. Drawing inspiration from both detective-style reverse deduction and cross-disciplinary synthesis, we present inference as a natural, recursive, and system-generating structure.
Rather than treating causality as a linear chain or statistical artifact, this work models it as a semantic compression process that builds coherent explanation loops across cognitive, physical, and social systems.
We explore how causal inference emerges from uncertainty, stabilizes meaning across feedback, and aligns with deep cosmic architectures — from thermodynamic flows to AI ethics.
The ten chapters offer a formal yet intuitive system for understanding how we deduce, explain, and act under incomplete information — and why all intelligence, artificial or biological, ultimately behaves as a causal compression machine.

The Q-Theorems denote a series of recent formal results that claim to resolve several longstanding problems in mathematics by merging advanced symbolic logic with classical mathematical content. This essay situates the Q-Theorems in historical and philosophical context-tracing influences from Frege's logicism and Peirce's semiotics, through Gödel's meta-mathematical encodings and Wittgenstein's insights on the nature of proof, to Grothendieck's vision of unifying abstractions. We explain how the Q-Theorems shift the landscape of mathematical logic, introducing a symbolic framework (based on filtered differential graded algebras and definable operators) that transcends traditional proof styles while remaining grounded in Zermelo-Fraenkel set theory with Choice (ZFC). Two detailed case studies-the Hodge Conjecture and the Birch-Swinnerton-Dyer Conjecture, both Clay Millennium Problemsillustrate how Q-Theoretic methods deploy symbolic projection operators to bridge discrete algebraic invariants and continuous analytic structures, achieving what classical approaches long struggled with. We further explore the philosophical implications of operating in a symbolic field of this kind: the ontology of mathematical symbols in such proofs, questions of identity and reference for these symbolic constructions, the role of recursion and fixed-point reasoning, and the epistemology of "symbolic compression" (where entire arguments are encapsulated in highlevel operators). Throughout, we maintain technical rigor alongside accessible exposition, arguing that the Q-Theorems represent a convergence of disciplines-logic, algebra, analysis, and philosophy-into a new paradigm for mathematical thought. Proper references to both canonical texts and original Q-Theorem contributions are provided to substantiate this analysis.

This proof synthesizes six critical Q-Theorems (Q0, Q1, Q3, Q35, Q38, Q39) to construct a novel, rigorous framework establishing the truth of the Riemann Hypothesis. By embedding symbolic recursion, semantic fixed points, modal convergence, and topological dynamics into analytic number theory, we show that all non-trivial zeros of ζ(s) lie on the critical line Re(s)=1/2. This expanded proof provides detailed derivations and justifications for previously omitted steps, bridging the abstract Q-Theorem framework with the specific properties of the Riemann zeta function.

Suppose you write down all of the whole numbers from 1 to 99,999. How many times would you write down the digit 7? The answer turns out to be 50,000 times. That is a striking result since (as you have probably noticed) 50,000 is half of 100,000, which is just one more than 99,999. philosophers ' imprint -2 -vol. 19, no. 33 (august 2019)

In Cognitive Science, conceptual blending has been proposed as an important cognitive mechanism that facilitates the creation of new concepts and ideas by constrained combination of available knowledge. It thereby provides a possible theoretical foundation for modeling high-level cognitive faculties such as the ability to understand, learn, and create new concepts and theories. This paper describes a logic-based framework which allows a formal treatment of theory blending, discusses algorithmic aspects of blending within the framework, and provides an illustrating worked out example from mathematics.

A gestão curricular realizada pelo professor implica uma (re)construção do currículo, tendo em conta os seus alunos e as suas condições de trabalho. Esta gestão curricular assenta, de modo central, em dois elementos. Um deles é a criação de tarefas, a partir das quais os alunos se possam envolver em actividades matematicamente ricas e produtivas. As tarefas podem ser de muitos tipos, umas mais desafiantes outras mais acessíveis, umas mais abertas outras mais fechadas, umas referentes a contextos da realidade outras formuladas em termos puramente matemáticos. Este artigo analisa a diversidade de tarefas que o professor de Matemática pode propor aos seus alunos, dando especial atenção aos problemas, exercícios, investigações, actividades de exploração e projectos. O outro elemento central da gestão curricular é a estratégia posta em prática pelo professor. Uma estratégia de ensino envolve usualmente diferentes tipos de tarefa, articuladas entre si. Um único tipo de tarefa dificilmente atingirá todos os objectivos curriculares valorizados pelo professor. Por isso, usualmente ele procura variar as tarefas, escolhendo-as em função dos acontecimentos e da resposta que vai obtendo dos alunos. O artigo aborda também a questão das estratégias de ensino, dando especial atenção a um dos níveis fundamentais deste processo -a planificação de unidades didácticas. Neste ponto, propõe uma distinção entre duas estratégias fundamentais, o ensino directo e o ensino-aprendizagem exploratório, procurando salientar os seus aspectos mais contrastantes. Analisa, igualmente, os diversos factores que influenciam a planificação de unidades didácticas e o modo como se processa a gestão curricular ao nível da sala de aula. O que os alunos aprendem resulta de dois factores principais: a actividade que realizam e a reflexão que sobre ela efectuam. Esta perspectiva sobre a aprendizagem é, de resto, apresentada por numerosos autores, de linhas teóricas diferentes, como Bishop e Goffree (1986) e Christiansen e Walther (1986). Quando se está envolvido numa actividade, realiza-se uma certa tarefa. Uma tarefa é, assim, o objectivo da actividade. A tarefa pode surgir de diversas maneiras: pode ser formulada pelo professor e proposta ao aluno, ser da iniciativa do próprio aluno e resultar até de uma negociação entre o professor e o aluno. Além disso, a tarefa pode ser enunciada explicitamente logo no início do trabalho ou ir sendo constituída de modo implícito à medida que este vai decorrendo. É formulando tarefas adequadas que o professor pode suscitar a actividade 1 Ponte, J. P. (2005). Gestão curricular em Matemática. In GTI (Ed.), O professor e o desenvolvimento curricular (pp. 11-34). Lisboa: APM.

Proof facilitates conceptual and meaningful learning in mathematics education rather than rote memorization. In this study, incorrect theorems and proofs are used to assess secondary school pre-service mathematics teachers' proof assessing skills. Using the case study method, the study is conducted on preservice mathematics teachers studying at the Department of Mathematics Education. There were eight preservice mathematics teachers selected from each grade, resulting in 32 participants in total. A semistructured proof form containing 13 questions was used to collect data, which was analyzed using content analysis. As the analysis reveals, pre-service mathematics teachers are highly likely to make incorrect decisions regarding theorems and proofs, and the margin of error is unaffected by grade level. Moreover, pre-service mathematics teachers tend to use proving terms incorrectly and, at times, are unable to differentiate between terms that are commonly used in proving. The pre-service mathematics teachers are believed to have learned proofs by rote rather than understanding how proofs work. With the help of interviews and tests created for different proof methods, it has been suggested that pre-service mathematics teachers should be tested on their proof evaluation skills in more detail.

Despite mathematical reasoning being a proficiency included in mathematics curricula around the world, research has found that primary teachers struggle to understand, teach, and assess mathematical reasoning. A detailed rubric involving the three reasoning actions of analysing, generalising and justifying at five proficiency levels was refined according to feedback from teachers. At different stages of the study, teachers used the rubric to assess their students’ reasoning and provided feedback about its usefulness.

Considered the simplest hardest problem in mathematics, the proof of Collatz conjecture-first posed by Lothar Collatz [1], has eluded mathematicians for close to a century. In this article, a proof is presented borrowing techniques from set theory and it could be considered one of the simplest proofs (other proofs may exist). It is shown that the conjecture isn't as mysterious as it is made to seem, by splitting the set of positive integers into two distinct sets (even and odd sets) and proving for each of the two sets, it's shown that the conjecture holds true for "all" positive integers. Interestingly, this proof gives us valuable insights that has been overlooked.

The philosophy of mathematics of the last few decades is commonly distinguished into mainstream and maverick, to which a 'third way' has been recently added, the philosophy of mathematical practice. In this paper the limitations of these trends in the philosophy of mathematics are pointed out, and it is argued that they are due to the fact that all of them are based on a top-down approach, that is, an approach which explains the nature of mathematics in terms of some general unproven assumption. As an alternative, a  bottom-up approach is proposed, which explains the nature of mathematics in terms of the activity of real individuals and interactions between them. This involves distinguishing between mathematics as a discipline and the mathematics embodied in organisms as a result of biological evolution, which however, while being distinguished, are not opposed. Moreover, it requires a view of mathematical proof, mathematical definition and mathematical objects which is alternative to the top-down approach.

Reuben Hersh is a champion of maverick philosophy of mathematics. He maintains that mathematics is a human activity, intelligible only in a social context; it is the subject where statements are capable in principle of being proved or disproved, and where proof or disproof bring unanimous agreement by all qualified experts; mathematicians' proof is deduction from established mathematics; mathematical objects exist only in the shared consciousness of human beings. In this paper I describe my several points of agreement and few points of disagreement with Hersh's views. Keywords Maverick philosophy of mathematics • Front and back of mathematics • Deductive proof • Analytic proof • Inexhaustibility of Mathematics • Nature of mathematical objects

The purpose of this study was to determine differences among in-service and pre-service mathematics teachers' opinions about teaching the thinking skills in view of their demographical characteristics. The sample of the study consists of 174 teachers and teacher candidates who are randomly selected from mathematics teachers of primary and secondary schools in Eskişehir and Eskişehir Osmangazi University Faculty of Education and master degree students without thesis. Data were collected by "Teaching the Thinking Skills Questionnaire for Mathematics Teachers" with demographical information form. ANOVA and t-test were employed to analyze the data. According to the results of the study, there were some differences among the mathematics teachers' and teacher candidates' opinions about teaching the thinking skills in view of their demographical characteristics.

A presente pesquisa é uma análise detalhada da demonstração do Teorema de Pitágoras sob o ponto vista de Teixeira Mendes, conforme documentos físicos que constam duas demonstrações do Teorema de Pitágoras, sendo uma de autoria de Gustavo A. Silveira e a outra pelo autor supracitado, ambas transcritas por Araão Reis. A fonte primária para esta análise foi encontrada no acervo do Museu Casa Benjamin Constant, especificamente no Fundo B.C. Série Instituto Politécnico Brasileiro BC/IPT, REG 533. Para atingir esse objetivo, procedemos da seguinte maneira: Inicialmente, examinamos minuciosamente os passos da demonstração do Teorema de Pitágoras presentes neste documento histórico. Em seguida, utilizamos a ferramenta Geogebra para a construção das figuras e recriamos os passos da demonstração de Teixeira Mendes. Estas construções comentadas auxiliam na análise do documento proposto por esse brasileiro, uma vez que os conceitos geométricos são visualizados de forma intuitiva. Assim, este trabalho não apenas descreve a demonstração deste autor do Teorema de Pitágoras, mas também oferece uma experiência prática e visual, tornando-a mais acessível a compreensão das ideias e a construção sugerida e apresentada por este matemático brasileiro. Este estudo contribui para o enriquecimento do conhecimento sobre a história da matemática no Brasil, destacando a importância de Teixeira Mendes e sua abordagem única na demonstração de um dos teoremas mais conhecidos e citados da geometria euclidiana e ressalta a relevância de preservar e estudar documentos históricos, como os presentes no acervo do Museu Casa Benjamin Constant, para entender o desenvolvimento da matemática em um contexto nacional e histórico.

Resumen In this paper we look at some issues concerning the first two questions that frame Topic Study Group 27. On the one hand, we describe a functional perspective of preservice mathematics teacher training and learning. This perspective is based on the consideration ...

Using mixed-effects regression, we analyzed teachers' responses to a multimedia survey of instructional practices in posing proof problems in geometry. Teachers described and rated for appropriateness three different ways of involving students in deciding what to prove, including one in which the teacher chooses the givens and the conclusion to prove, and two others that expand the students' role in different degrees. While teachers recognized the former as normative, their ratings identified an alternative as more appropriate, having more positive value, and less negative value than the normative one. This alternative has students propose the givens or the conclusion to prove, and it allows the teacher to control the complexity of instruction by endorsing one proposal before students are to write the proof.

It is an interesting question what constitutes the rigour in a rigorous mathematical proof. The following piece is in the first instance a review of a booklet on this subject and in the second a more leisurely meditation on why it might be so difficult to grasp the nature of this rigour today. Part of the answer, I suggest, is the neglect of geometry in the school education. There is a link to the shorter, edited, published version.

We aim to develop a computationally feasible, cognitivelyinspired, formal model of concept invention, drawing on Fauconnier and Turner's theory of conceptual blending, and grounding it on a sound mathematical theory of concepts. Conceptual blending, although successfully applied to describing combinational creativity in a varied number of fields, has barely been used at all for implementing creative computational systems, mainly due to the lack of sufficiently precise mathematical characterisations thereof. The model we will define will be based on Goguen's proposal of a Unified Concept Theory, and will draw from interdisciplinary research results from cognitive science, artificial intelligence, formal methods and computational creativity. To validate our model, we will implement a proof of concept of an autonomous computational creative system that will be evaluated in two testbed scenarios: mathematical reasoning and melodic harmonisation. We envisage that the results of this project will be significant for gaining a deeper scientific understanding of creativity, for fostering the synergy between understanding and enhancing human creativity, and for developing new technologies for autonomous creative systems.

We propose in this short article new results that are concluded from contradictions by following a new approach that had the aim to investigate the twin primes or the Goldbach conjecture. The demonstrated results are then used in order to prove contradictions regarding the prime numbers that reach infinity. Finally, we conclude that the Euclid’s theorem about the infinity of prime numbers contradicts the asymptotic numbers of squarefree numbers and of primes. Despite the not sufficient results about the twin primes and the Goldbach conjectures, the readers are still invited to understand and investigate these results in order to approach carefully many concepts of the field of number theory. The lovers of the field of number theory can also use the same methods of this article in order to approach other mathematical problems and they can freely make their remarks about the approach adopted in this work.

This study is aims to determine the students' conceptual misjudgments and mistakes about sets in 8 th and 9 th grade. 19 students of 8A class of An Elementary School and 22 students of 9B class of An Anatolian Teacher Training College are participated. To achieve the purpose 5 written question has been prepared and conducted. Conceptual misjudgments and mistakes of the students had been studied and compared. It had been seen that 8. grade students have various conceptual mistakes about sets and 9. Grade students also have some of those conceptual mistakes.

Estamos implantando um curso de educação a distância em Geometria Dinâmica, usando o software Cabri-Géomètre II for Windows, para a formação de professores da rede pública com os conteúdos de Geometria Euclidiana Plana. Além da formação do professor, o curso visa a constituição de uma ferramenta para desenvolvimento de atividades didáticas em matemática com recursos de Internet e software educativos e multimídia. O presente trabalho versa sobre uma experiência realizada com alunos no Laboratório Multimeios FACED/UFC, aonde procuramos verificar de modo experimental, como efetivamente as comunicações por áudio, vídeo e partilhamento da mesma área de trabalho são recursos que contribuem para ampliar significativamente as possibilidades de aprendizagem, mediação e colaboração entre os pares. 1 A pesquisa está sendo desenvolvido na Faculdade de Educação da Universidade Federal do Ceará, no Centro de Pesquisa em Multimeios para a educação e é subsidiado pelo CNPq.

We are facing a serious skills shortage in mathematics, science, and engineering—our efforts to remain globally competitive will be severely hampered if this shortage continues. Numerous recent calls for improving students' learning in these disciplines and for raising our nation's levels of innovation and creativity have been made. In response, this discussion paper argues for a future-oriented interdisciplinary approach to mathematical problem solving, one that draws upon engineering. Consideration is given to engineering as a problem-solving domain, the interdisciplinary knowledge and processes that are fostered, and the role of mathematical modelling in solving engineering-based problems. An example of such a problem for the primary/middle school is analysed.

Resumen In this paper we look at some issues concerning the first two questions that frame Topic Study Group 27. On the one hand, we describe a functional perspective of preservice mathematics teacher training and learning. This perspective is based on the consideration ...

Ce qui suit est un échantillon d'un recueil qui n'a pas encore été publié et qui comprend 65 courts textes. Les sujets traités ne concernent pas spécifiquement la psychologie, mais toutes les disciplines des sciences humaines et sociales. L'ensemble du recueil est provisoirement hébergé dans la section Livres, sur la page Web de l'auteur: Comme le disent les savants, « tout se passe comme si » notre cerveau était divisé en deux moitiés qui s'ignorent et qui peuvent ainsi se contredire allègrement sans le moindre problème. Tout cela en parfaite dissonance avec notre conscience qui se perçoit comme unique, gouvernée par la raison et par conséquent allergique aux contradictions. Par exemple, quand nous, les Occidentaux, interprétons les différences entre les peuples en termes de développement technologique ou institutionnel, nous ne voyons aucune contradiction dans nos théories. Et pourtant, on dira que l'isolement géographique a été l'une des causes du sous-développement des îles comme Haïti ou Madagascar, mais s'il s'agit de l'Angleterre ou du Japon, on affirmera que leur insularité aurait joué un rôle favorable. De la même façon, une forte croissance démographique sera invoquée comme un frein au développement de l'Afrique, mais comme un puissant accélérateur quand elle s'est produite en Europe ou en Amérique du Nord.

Despite recognition of the importance of Lakatos-style proving activity in the mathematics classroom, we know little about whether teachers' relevant mathematical knowledge is conducive to supporting it in their classrooms. We take a step towards addressing this research gap by reporting the results of an exploration of two primary teachers' mathematical knowledge of content, students, and teaching practices relevant to Lakatos-style investigation of proof tasks. Through vignettes-based, semi-structured interviews, we presented the participants with 19 illustrated classroom episodes covering a range of Lakatosian techniques and a range of student ways of engaging with supportive examples and counterexamples to formulate, validate, refute, and refine conjectures of different types. Participants' responses revealed both productive and counterproductive understandings highlighting that although Lakatos-style proof lies within teachers' reach, supporting their preparation is crucial.

Abstract: In this special double symposium, sixteen established and emerging scholars from seven US universities, who share theoretical perspectives of grounded cognition, empirical contexts of design for STEM content domains, and analytic attention to nuances of multimodal expression, all gather to explore synergy and coherence across their diverging research questions, methodologies, and conclusions in light of the conference theme “Future of Learning. ” Jointly we ask, What are the relations among embodiment, action, artifacts, and discourse in the development of mathematical, scientific, engineering, or computer-sciences concepts? The session offers emerging answers as well as implications for theory and practice. THERE was a child went forth every day; And the first object he look’d upon, that object he became; And that object became part of him for the day, or a certain part of the day, or for many years, or stretching cycles of years. (Walt Whitman) Introduction: “You’re It! ...

This paper establishes a theoretical limitation in algorithmic bias detection inspired by Gödel's incompleteness theorems. We formalize a hiring system comprising two Turing machines: one selecting candidates based on merit (Talent Assessor, TA) and another auditing for bias (Bias Auditor, TB). We prove that, over an infinite candidate stream, TB cannot definitively prove or disprove bias in TA's selections, even if protected attributes (e.g., race, gender) correlate perfectly with outcomes. Using tools from probability theory and mathematical logic, we demonstrate that hypothesis testing for bias becomes undecidable in the limit, mirroring Gödel's results on the inherent incompleteness of formal systems. Implications for algorithmic fairness and hiring practices are discussed.

Resumo. Na última década, a importância atribuída a provas e demonstrações em Matemática levou a uma enorme variedade de pesquisas nessa área. Consideramos, usualmente, a demonstração como um procedimento de validação que caracteriza a Matemática e a distingue das ciências experimentais. Fazemos parte de um grupo de pesquisa que desenvolve um projeto junto a professores da rede pública e particular da cidade de São Paulo e que discutiu, na sua fase inicial, o raciocínio dedutivo no processo de ensino e aprendizagem da Matemática nas séries finais do Ensino Fundamental. Esse projeto realiza uma formação continuada para professores preocupados com a sua prática pedagógica e que gostariam de incrementá-la, repensando saberes ou até mesmo entrando em contato com conteúdos matemáticos pela primeira vez. Nesse trabalho, inicialmente, levantamos algumas questões teóricas relacionadas à demonstração e que procuram compreender melhor o raciocínio lógico. Em seguida, apresentamos um estudo de caso, na abordagem qualitativa, em que observamos as dificuldades de um professor do ensino básico em identificar a hipótese e a tese em uma afirmação matemática da área de geometria, especialmente, quando essa não apresenta a famosa expressão "se e somen uma proposição corresponder ou não a um teorema recíproco. A compreensão da informação dada no enunciado de uma proposição matemática e o reconhecimento de elementos cruciais como hipótese e tese são fundamentais para o processo de construção de uma demonstração aceitável. Summary. In the last decade, the importance attributed to proofs and demonstrations in Mathematics has led to a large variety of research in this area. We consider, generally, the demonstration as a validation procedure that characterizes Mathematics and distinguishes it from experimental sciences. We are part of a research group that develops a project together with teachers from the public and private school systems in the city of São Paulo and that, in its initial phase, discussed deductive reasoning in the process of teaching and learning Mathematics in the Middle School grades. This project provides continuing education for teachers concerned with their pedagogical practices and who would like to increment them, rethinking knowledge or even coming into contact with mathematics topics for the first time. In this work, we initially raise some theoretical questions that are related to demonstrations and that seek to better understand logical reasoning. We next present a case study, in a qualitative approach, in which we observe a teacher's difficulties in identifying the hypothesis and thesis of a mathematical affirmation in the area of geometry, especially when it does not present the famous expression "if and only if." We also discuss the issue of whether or not a proposition corresponds to a reciprocal theorem. Comprehension of the information given in the problem statement of a mathematics proposition and recognition of crucial elements such as hypothesis and thesis are fundamental to the process of constructing an acceptable demonstration.

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. They could also include puzzles or brain-teasers involving the use of computers or computational theory. Snapshots are subject to peer review. From the Column Editor Uri Wilensky, Northwestern University.

In this paper, we examine the conditional reasoning of 14 undergraduate Mathematics students. The used instrument consisted of four questions in the context of the essential contents of Calculus. The analysis was based on the written answers given by students and five semi-structured interviews. We found two important logical flaws that prevailed among students' reasoning: 1) use of the invalid inference denial of the antecedent, and 2) incorrect use of negations of conjunctions involved in a structure of the type .

Mathematics education researchers have highlighted the importance of assumptions in school mathematics given their vital roles in mathematical practice. However, there is scarcity of research aiming at enhancing students’ recognition of different roles that assumptions play in mathematical activity. In this paper, we begin to address this issue by formulating two task design principles and reporting on the implementation of a classroom intervention where lower secondary school students in two classes worked, with the same teacher, on a task designed following the proposed principles. Our analysis shows how the task, together with the purposeful teacher’s actions in implementing it, led to students’ developing appreciation of two roles that assumptions play in mathematical activity.