Immediate consequences operator on generalized quantifiers

The semantics of a multi-adjoint logic program is usually defined through the immediate consequences operator TP. However, the definition of the immediate consequences operator as the supremum of a set of values can provide some problem when imprecise datasets are considered, due to the strict feature of the supremum operator. Hence, based on the flexibility of generalized quantifiers to weaken the existential feature of the supremum operator, this paper presents a generalization of the immediate consequences operator with interesting properties for solving the aforementioned problem. © 2022 The Author(s

Analyzing Mathematicians\u27 Concept Images of Differentials

The differential is a symbol that is common in first- and second-year calculus. It is perhaps expected that a common mathematical symbol would be interpreted universally. However, recent literature that addresses student interpretations of differentials, usually in the context of definite integration, suggests that this is not the case, and that many interpretations are possible. Reviews of textbooks showed that there was not a lot of discussion about differentials, and what interpretations there were depended upon the context in which the differentials were presented. This dissertation explores some of these issues. Since students may not have the experience necessary to build their own interpretations totally free of their instructors’ influences, I chose to interview experienced mathematicians for their differential interpretations. Most of the current literature involves the differential within the context of definite integrals; my work expands on this literature by exploring additional expressions that contain differentials. The goal was to build a dataset of multiple instructors’ interpretations of multiple differentials to see how uniform those interpretations were. Initial interviews discussing five expressions which contained differentials, three contexts in which most of these expressions were used, and auxiliary questions that asked the meaning of “differential,” the differences between and , and the interpretation of phrases used to describe infinitely small quantities were conducted with seven expert mathematicians from a large research university. By analyzing the responses given by these mathematicians, two lists of themes were created: one based on remarks that address the quality of the differential directly, and one based on remarks that address one’s feelings about differentials. In addition, for the responses that address differentials directly, a flowchart was created to guide each of these responses to its proper theme. After the creation of these lists, three more mathematicians were interviewed to ensure that the theme lists would still be valid outside of the interviews used to create them. Not only was no overall formal concept image for the differential found, but many different and sometimes contrasting themes were found within each interview subject’s personal concept image. A framework for categorizing the multiple conceptualizations that were found for the differentials themselves was created, as well as a beginning list of ancillary themes that address possible thoughts about and uses of differentials. The dissertation concludes with a list of possible teaching implications that might arise from the existence of multiple differential conceptualizations, as well as some suggested future research that might expand upon this work

Representações em situações problemáticas que envolvem inequações do 1.º grau a uma incógnita : um estudo com alunos do 9.º ano de escolaridade

Relatório da Prática de Ensino Supervisionada, Ensino da Matemática, Universidade de Lisboa, 2013O objetivo deste trabalho é perceber se alunos do 9.º ano compreendem e sabem usar os diferentes tipos de representações na resolução de situações problemáticas integradas no estudo de inequações. Para tal, formulei as seguintes questões: Quais são os principais tipos de representações usados pelos alunos na resolução de situações problemáticas que envolvem inequações do 1.º grau? Quais são os principais erros e dificuldades que os alunos revelam na conversão e no tratamento de representações de situações problemáticas que envolvem inequações do 1.º grau? A investigação decorreu no âmbito da minha prática letiva supervisionada durante o 3.º período do ano letivo de 2012/2013 numa turma do 9.º ano de uma escola, em Lisboa, tendo abordado o tema Álgebra, o tópico Inequações e o subtópico Inequações do 1.º Grau a uma Incógnita. A metodologia de investigação de natureza interpretativa, recorre a dados quantitativos e qualitativos. Participaram todos os alunos da turma, tendo selecionado dois deles para aprofundamento do estudo. Na recolha dos dados utilizei os seguintes instrumentos: a observação de aulas, alguns documentos da escola, um diário de bordo, as produções escritas de todos os alunos da turma e uma entrevista semiestruturada, gravada em áudio, aos dois alunos da turma, já mencionados, que obtiveram o melhor desempenho escolar na disciplina de Matemática no 1.º período do referido ano letivo. Os resultados do estudo revelam que os alunos passaram progressivamente do uso predominante da representação numérica para a representação algébrica. Na maioria das situações problemáticas, os alunos apresentaram dificuldades na conversão da linguagem natural para a linguagem algébrica, principalmente na escolha do sinal de desigualdade apropriado para cada caso. No tratamento das inequações, os alunos cometeram erros essencialmente na aplicação do 2.º princípio de equivalência e na construção do intervalo que represente adequadamente o respetivo conjunto-solução. As conclusões inferidas pela análise das produções escritas e da entrevista realizada aos dois alunos selecionados estão de acordo às já proferidas para toda a turma.The aim of the present investigation is to study whether the 9th year students understand and know how to use the different types of representations in solving problematic situations involving inequalities. Thus, I have formulated the following questions: Which are the main types of representations used by students in solving problematic situations involving inequalities of first degree? Which are the main mistakes and difficulties that students exhibit in the conversion and treatment of representations of problematic situations involving inequalities of first degree? The research occurred in the my teaching practice supervised during the third .period of the school year 2012/2013 in school in Lisbon, having discussed the theme Algebra, the topic Inequalities and the subtopic Inequalities of first degree. In data collection used the following instruments: classroom observation, some documents from the school, written work produced by all students during classes, and a semi-structured interview, recorded on audio, given by two students who obtained the best mathematical performance on in the first third period of the school year. The results of the study reveal that students were passing progressively from the predominant use of the numerical representation for the algebraic representation. Most of the problematic situations, students presented difficulties in the conversion of the natural language to the algebraic language, especially in the choice of appropriate inequality sign for each case. In the treatment of inequalities, the students made mistakes in the application of the 2nd equivalence principle and construction of the interval that adequately represents the respective solution set. The conclusions inferred by analyzing the written productions and interview the two students are according to those already referred to the whole class

Development of open education materials in mathematics for secondary school in the topic "Algebra expressions and equations"

Diplomdarbā "Tālmācības materiālu izstrāde matemātikā vidusskolā tēmai „Algebriskas izteiksmes un vienādojumi"" ir sagatavotas 8 nodarbības nekātienes klasēm Rīgas 14. vakara (maiņu) vidusskolā, ar mērķi nodrošināt materiālu pieejamību ārpus skolas telpām. Sagatavotās nodarbības tiks ievietotas skolas uzturētajā, slēgtajā Moogle platformā. Katra nodarbība ietvar teorijas izklāstu, uzdevumus un to atrisinājumu. Katras nodabības sākumā ir nprādīts,kas jāprot pēc nodarbības apguves,kā arī noradīti nodarbības atslēgvārdi un tiešsaistes materiāli, kur pieejama papildus informācija un uzdevumi par konkrēto tēmu. Atslēgvārdi: algebriskas izteiksmes, algebriski vienādojumi, tālmācībaDiploma thesis "Development of distance educatioan materials for the secondary school theme "Algebra expressions and equasition" included development of eight distance learning lessons for Riga Secondary school No.14. The school provides evening education programmes for students who are not able to attend the full-day-schooling education institutions. The lessons will be published in the schools' online platform,thus accessible to each of the students without time or place limitations. Each of the lessons include the following: aims of the lesson, keywords of the lesson, links to other relevant online learning materials, explanations of theory, exercises, explanations and solutions. Keyword of the diploma thesis: algebra expressions, algebra equations, distance learning

Ãlgebra ou geometria? Vamos a questÃo !

CoordenaÃÃo de AperfeÃoamento de Pessoal de NÃvel SuperiorO presente trabalho tem como tema: Ãlgebra ou Geometria? Vamos à questÃo! Esse estudo discute, principalmente, a relaÃÃo entre essas duas Ãreas distintas do ponto de vista curricular, no ensino bÃsico. O objetivo à mostrar como à tÃnue, e ao mesmo tempo frutÃfera, a fronteira que separa a Ãlgebra bÃsica e a Geometria Plana e, como o professor deve investigar esta fronteira logo nas primeiras sÃries do ensino mÃdio, mesmo antes de apresentar a Geometria AnalÃtica. Este estudo està organizado em forma de capÃtulos, abordando as seguintes temÃticas em ordem: o porquà do ensino da MatemÃtica, o mÃtodo utilizado na resoluÃÃo de problemas como processo, o pressuposto teÃrico, dando destaque à semelhanÃa entre triÃngulos, ao teorema de PitÃgoras, Ãs leis dos Senos e Cossenos, ao teorema de Ptolomeu, e à lista de problemas propostos e uma discussÃo epistemolÃgica dos problemas propostos. O trabalho foi à luz das propostas teÃricas de Elon Lages Lima, Terence Tao e de Paulo Freire, este Ãltimo um mestre da Pedagogia. A pesquisa foi feita de modo exploratÃrio e bibliogrÃfico, de carÃter qualitativo. Por fim, o estudo pretende deixar claro que a divisÃo curricular das aulas de MatemÃtica, no ensino bÃsico, em Ãlgebra e Geometria à apenas um divisÃo curricular e nÃo deveria afetar a visÃo de que todos os conteÃdos estudados fazem parte de um todo perfeitamente coerente; tendo em vista que, estando diante de um problema de matemÃtica, estudantes e professores podem lanÃar mÃo tanto de ferramentas da Ãlgebra quanto de resultados da geometria para resolvÃ-lo.The present work has as its theme: Algebra or Geometry? Let the question! This study discusses mainly the relationship between these two distinct areas of the curriculum perspective, basic education. The goal is to show how tenuous, and at the same fruitful time, the border that separates the basic algebra and plane geometry, and how the teacher should investigate this boundary in the very first year of high school, even before submitting Analytic Geometry. This study is organized in the form of chapters, covering the following topics in order: why the mathematics teaching, the method used in problem solving as a process, the theoretical assumption, highlighting the similarity of triangles, the Pythagorean theorem, the laws of sines and cosines, the Ptolemy's theorem, and the list of proposed issues and an epistemological discussion of the proposed problems. The work was in the light of theoretical proposals Elon Lages Lima, Terence Tao and Paulo Freire, the latter a master of pedagogy. The survey was conducted exploratory and bibliographic way, qualitative character. Finally, the study aims to clarify that the curriculum division of mathematics classrooms in primary school in Algebra and Geometry is only one curriculum division and should not affect the view that all the contents studied are part of a whole perfectly consistent; given that, being on a math problem, students and teachers can make use of both tools of algebra as geometry of the results to solve it