In tropical algebras we substitute min or max for the typical addition and then substitute addition for multiplication. A dot product representation of a graph assigns each vertex of the graph a vector such that two edges are adjacent if and only if the dot product of their vectors is greater than some chosen threshold. The resultS of creating dot product representations of graphs using tropical algebras are examined. In particular we examine the tropical dot product dimensions of graphs and establish connections to threshold graphs and the threshold dimension of a graph.

In many mathematical problems, we are expected to compare the interesting quantities. Thus, the use of well-known inequalities will be required. Techniques of using these inequalities to solve inequality problems vary from problem to problem. In this paper, we will introduce commonly used well-known inequalities in high school mathematical contests and discuss the solving strategies for inequality problems.

<p>The purpose of the thesis was to investigate and evaluate existing algorithms for division of complex numbers. The investigation should include implementation of a few suitable algorithms in VHDL. The main application for the divider is compensation for fading in a baseband processor.</p><p>Since not much public research is done within the area of complex division in hardware, a divider based on real valued division was designed. The design only implements inversion of complex numbers instead of complete division because it is simpler and the application does not need full division, thus the required chip size is reduced.</p><p>An examination of the different kinds of algorithms that exists for real valued division was done and two of the methods were found suitable for implementation, digit recurrence and functional iteration. From each of the two classes of algorithms one algorithm was chosen and implemented in VHDL. Two different versions of the inverter were designed for each method, one with full throughput and one with half throughput. The implementations show very similar results in terms of speed, size and performance. For most cases however, the digit recurrence implementation has a slight advantage.</p>

Syftet med studien är att bidra till ökad förståelse av och fördjupad kunskap om taluppfattningens och skriftliga räknemetoders betydelse för hinder i elevers matematikutveckling, särskilt avseende addition och subtraktion, samt undersöka hur lärare arbetar med dessa områden för att förebygga och möta hinder för matematikutveckling. Elevers matematikkunskaper sjunker och på senare år har brister i taluppfattning uppmärksammats som en möjlig orsak. Denna studie med kvalitativ ansats har intervjuer och skriftliga dokument som datainsamlingsmetod. Hur uppfattar nio lärare som undervisar i år 1-6 nödvändiga kunskaper i taluppfattning för att hantera skriftliga räknemetoder i addition och subtraktion och vilka förklaringar till brister lyfter de. Vilka verktyg används för att få kännedom om elevers kunskaper i matematik vad gäller taluppfattning och skriftliga räknemetoder? Av resultatet av studien framkommer att det finns variationer i uppfattningar om nödvändiga kunskaper och undervisning om taluppfattning och skriftliga räknemetoder. Resultaten på Nationella prov vad gäller de båda studerade områdena visar på ett bättre resultat då det gäller taluppfattning jämfört med skriftliga räknemetoder. Detta kan bero på den komplexitet som det sociala samspelet mellan olika strukturer i samhället innebär.

The purpose of the thesis was to investigate and evaluate existing algorithms for division of complex numbers. The investigation should include implementation of a few suitable algorithms in VHDL. The main application for the divider is compensation for fading in a baseband processor. Since not much public research is done within the area of complex division in hardware, a divider based on real valued division was designed. The design only implements inversion of complex numbers instead of complete division because it is simpler and the application does not need full division, thus the required chip size is reduced. An examination of the different kinds of algorithms that exists for real valued division was done and two of the methods were found suitable for implementation, digit recurrence and functional iteration. From each of the two classes of algorithms one algorithm was chosen and implemented in VHDL. Two different versions of the inverter were designed for each method, one with full throughput and one with half throughput. The implementations show very similar results in terms of speed, size and performance. For most cases however, the digit recurrence implementation has a slight advantage.

Orientador: Antônio Carlos Patrocinio / Dissertação (mestrado profissional) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica / Made available in DSpace on 2018-08-25T12:22:05Z (GMT). No. of bitstreams: 1 Pereira_JaksonDaCruz_M.pdf: 1462332 bytes, checksum: 393b1f36be156bf1cdedb16da1cc3fcd (MD5) Previous issue date: 2014 / Resumo: O presente trabalho se dedica ao estudo das médias aritmética, geométrica e harmônica. Inicialmente, definimos cada uma das médias e trabalhamos suas aplicações através da resolução de problemas. Posteriormente destacamos as desigualdades entre as médias e suas aplicações / Abstract: This dissertation is dedicated to the study of the arithmetic, geometric and harmonic means. Initially, we defined each of the means and applied its uses through problem resolution. Afterwards, we gave emphasis to the inequalities between the means and its uses / Mestrado / Matemática em Rede Nacional / Mestre em Matemática em Rede Nacional

Med denna litteraturstudie var syftet att sammanställa forskning kring laborativt material och dess effekt på aritmetikundervisningen för elever i de lägre årskurserna. Forskningsunderlaget samlades in via databasen ERIC Ebsco.  Resultatet påvisar en övervägande positiv effekt på aritmetikinlärningen för elever och definierar aspekter för hur dessa positiva effekter kan nås. Resultatet indikerar även att lärare har blandad kunskap kring manipulering av laborativt ma-terial och kring materialets användning i undervisningen. För en god undervisning behöver lärare få tillgång till mer kunskap kring laborativt material. Detta ger elever en ökad möjlighet att nå avsikten med manipulering av laborativt material, nämligen att se de bakomliggande matematiska principerna. / The purpose of this literature review was to compile research on manipulative materials and its effect on arithmetic teaching for students in the lower grades. Research data was collected through the ERIC Ebsco database. The results show a predominantly positive effect on arith-metic learning for students and define aspects for how these positive effects can be achieved. The result also indicates that teachers have mixed knowledge about the manipulation of ma-nipulative materials and the use of the materials in teaching. For a good teaching situation, teachers need access to more knowledge about manipulative materials. This gives students an increased opportunity to reach the intention of manipulating manipulative materials, namely to see the underlying mathematical principles.

Η μετάβαση των μαθητών από την Πρωτοβάθμια στη Δευτεροβάθμια Εκπαίδευση κατά γενική ομολογία, δημιουργεί πολλές δυσκολίες στους μαθητές, οι οποίες είναι ιδιαίτερα έντονες στο μάθημα των Μαθηματικών. Αυτό συμβαίνει γιατί οι παραπάνω έρχονται αντιμέτωποι με έναν καινούριο χώρο όπου έχει έναν διαφορετικό τρόπο σκέψης και γραφής από αυτόν που είχαν συναντήσει στην Αριθμητική κατά τη φοίτηση τους στις τάξεις του Δημοτικού. Ο χώρος αυτός δεν είναι άλλος από τον ‘όμορφο’ χώρο της Άλγεβρας. Οι μαθητές λοιπόν, στο μάθημα της Άλγεβρας έρχονται αντιμέτωποι με σωρεία καινούριων πληροφοριών παρατηρώντας πολλές διαφορές αλλά και αρκετές ομοιότητες με τις γνώσεις που είχαν λάβει από την Αριθμητική, γεγονός που πολλές φορές τους δημιουργεί σύγχυση. Ιδιαίτερα έντονες είναι οι δυσκολίες τους στην κατανόηση βασικών εννοιών όπως της μεταβλητής αλλά και στην επίλυση μιας πρωτοβάθμιας εξίσωσης που παρουσιάζονται στην Άλγεβρα. Η παρούσα διπλωματική εργασία αποτελείται από μια μελέτη της υπάρχουσας βιβλιογραφίας και των ερευνών που έχουν γίνει προς αυτή τη κατεύθυνση από πλήθος Ελλήνων και ξένων ερευνητών καθώς και παράθεση ερωτηματολογίου σε μαθητές Γ’ Γυμνασίου, εντοπισμός των βασικών λαθών τους και συνήθων συγχύσεων. Αναλυτικότερα, η δομή της εργασίας έχει ως εξής: • Στο κεφάλαιο 1 γίνεται μια ιστορική αναδρομή για το πώς φτάσαμε από τα πρώτα πρώιμα μαθηματικά, στην χρήση αγνώστων και στην επίλυση εξισώσεων έως τα σύγχρονα χρόνια καθώς και τη μετάβαση από την Αριθμητική στην Άλγεβρα. • Στο κεφάλαιο 2 επισημαίνεται η δυσκολία που συναντούν οι μαθητές από τη μετάβασή τους στη δευτεροβάθμια εκπαίδευση. Παραθέτονται ενδεικτικά οι έρευνες ομάδων ερευνητών Ελλήνων και μη και επισήμανση συγκεκριμένων κοινών χαρακτηριστικών. Στο σημείο αυτό περιγράφουμε το σκοπό της έρευνας μας καθώς και την επιλογή των μαθητών που παραθέτουμε το ερωτηματολόγιο, τη μέθοδο συλλογής και καταγραφής των αποτελεσμάτων. • Στο κεφάλαιο 3 παραθέτουμε το ερωτηματολόγιο που δώσαμε στους μαθητές χωρισμένο σε Α και Β ομάδα και εξηγούμε το στόχο κάθε ερώτησης που θέσαμε. Κατόπιν, αναλύουμε εκτενώς τις απαντήσεις των παιδιών, παραθέτοντας και αυτούσιους διαλόγους που είχαμε μαζί τους. Εντοπίζουμε έτσι τα πιθανά λάθη και παρερμηνεύσεις που αναμέναμε αλλά και ό,τι επιπλέον προέκυψε από τις απαντήσεις τους. • Τέλος, στο κεφάλαιο 4 καταστρώνουμε ένα διδακτικό σχέδιο βασισμένο στα λάθη των μαθητών και τις λοιπές παρατηρήσεις που είδαμε νωρίτερα. Δίνουμε ένα πλάνο διδασκαλίας της άλγεβρας στις μαθητικές αίθουσες, με ποιους τρόπους θα πρέπει να μεταλαμπαδεύουμε τις γνώσεις και την «όρεξη» μας για την άλγεβρα και με ποια τεχνολογικά μέσα. Έπειτα συνδέουμε την διδασκαλία αυτή με ένα ερωτηματολόγιο ελέγχου επιτυχίας της προηγούμενης διαδικασίας. Κλείνουμε με τα συμπεράσματα της έρευνάς μας. / -

Orientador: Orly Zucatto Mantovani de Assis / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Educação / Made available in DSpace on 2018-08-15T18:49:59Z (GMT). No. of bitstreams: 1 Molinari_AdrianaMariaCorder_D.pdf: 14984989 bytes, checksum: 27a2ea55ce633927a07aeebe2a870560 (MD5) Previous issue date: 2010 / Resumo: O objetivo desta pesquisa foi verificar como as crianças de 4º e 5º anos representam graficamente procedimentos de solução de problemas aritméticos de divisão por quotas. Fundamentado na teoria de Jean Piaget, este estudo descreve o processo de construção da operação aritmética de divisão e foi realizado com vinte alunos matriculados no Ensino Fundamental I de uma escola privada, localizada no interior do estado de São Paulo. Participaram dez alunos do 4º ano e dez do 5º, com idade entre 9 e 11 anos. Para verificar as representações dos estudantes, aplicaram-se provas aritméticas de divisão, compostas de seis problemas de divisão por quotas no total, distribuídos em duas sessões: a Prova de Multiplicação e Divisão Aritmética, cuja meta foi avaliar o nível da psicogênese da noção de multiplicação e de divisão dos estudantes, e a Entrevista, cuja meta foi verificar a explicação dos educandos aos procedimentos de solução empregados, bem como a noção de divisão construída. Do ponto de vista da psicogênese da noção do operador multiplicativo, os estudantes incluíram-se nas condutas III, IV, ou em transição entre as condutas III e IV, revelando estarem bem desenvolvidos nessa noção; porém verificou-se que somente 4 dos 20 estudantes, em ambos os anos de escolaridade, estavam de posse do operador multiplicativo; por outro lado, apesar de não apresentarem tal noção construída, a maioria deles demonstrou conhecer as técnicas convencionais de solução de problemas. Os resultados revelaram o emprego de uma diversidade de procedimentos de solução de problemas, que variou do desenho (forma mais elementar de representação) até o algoritmo canônico da divisão (forma mais avançada, do ponto de vista da convenção). A análise qualitativo-quantitativa do estudo mostrou uma variação dos procedimentos de solução em ambos os anos de escolaridade; evidenciou também a inexistência de uma relação necessária entre a complexidade do procedimento de solução e o ano de escolaridade, uma vez que procedimentos mais avançados foram encontrados entre estudantes do 4º ano, assim como procedimentos mais elementares, entre estudantes do 5º ano. / Abstract: The goal of this research was to verify how four and five-year-old children graphically represents the procedures of arithmetic quota division problem-solving. Based on the theory by Jean Piaget, this work was accomplished with twenty students enrolled in Elementary School in a private school located in the countryside of São Paulo state. Ten students of 4th grade and 10 students of 5th grade took part in the study, from nine to eleven years old. To verify the students representation, it was applied arithmetic division tests, consisted of six problems of quota division, distributed into two sessions: the Multiplication and Arithmetic Division Test, which goal was to evaluate the level of psychogenesis multiplication and division of students, and the Interview, which goal was to verify the students explanation for the solving procedures applied, as well as the division idea that was built. From the standpoint of the conception of the multiplicative operator psychogenesis, the students were included in the conducts III, IV or in transition between conducts III and IV, or in transition between conducts III and IV, revealing to be well developed in this conception; yet, it was found that only four of the twenty students, in both School grades, possessed the multiplicative operator; however, in despite of they did not reveal this idea built, most of them proved to know the conventional techniques of problem-solving. The results revealed a diversification in the use of the procedures for problem-solving, which alternated from the drawing (most elementary form of representation) to the canonical division algorithm (advanced way, from the convention point of view). The qualitative / quantitative study showed a variation of the solving procedures in both School grades; it also became evident the inexistence of a necessary relationship between the complexity of the solving procedure and the School grade, once the advanced procedures were found among students from 4th grade, as well as more elementary procedures, among students from 5th grade. / Doutorado / Psicologia, Desenvolvimento Humano e Educação / Doutor em Educação

Orientador: Marcia Regina Ferreira de Brito Dias / Tese (doutorado) - Universidade Estadual de Campinas, Faculdade de Educação / Made available in DSpace on 2018-08-19T10:04:10Z (GMT). No. of bitstreams: 1 Quintiliano_LucianedeCastro_D.pdf: 2480214 bytes, checksum: 3c100143a02fb8891af9cf1d3408a2bc (MD5) Previous issue date: 2011 / Resumo: O presente estudo teve como objetivo verificar a existência de relações entre os estilos cognitivos e as estratégias de solução de problemas, os estilos cognitivos e as variáveis gênero e série, os estilos cognitivos e o desempenho dos estudantes na solução dos problemas, bem como o desempenho e o tipo de estratégia utilizada na solução dos problemas. Para tanto, foi solicitado à 168 estudantes da primeira, segunda e terceira série do Ensino Médio provenientes de duas escolas públicas que respondessem à um questionário informativo, uma escala tipo Likert para categorizar os estilos cognitivos, e uma prova matemática, abordando os conceitos e procedimentos algébricos e aritméticos. Através da análise em relação aos estilos cognitivos deste estudo, verificou-se a predominância da Divergência sobre a Convergência de pensamento. E a partir da análise das médias das variáveis, estilo cognitivo e gênero, as médias encontradas indicaram que as mulheres, em comparação aos homens, apresentaram um maior predomínio nos estilos Reflexividade e Serialista e menor nos estilos cognitivos Impulsividade e Independência de campo. Ambos os gêneros apresentaram predominância no estilo Dependente de campo, Serialista, Reflexividade. Verificou-se ainda a existência de diferenças significativas nos estilos cognitivos, quando considerada a série de estudo, reveladas entre os grupos de acordo com o estilo cognitivo Convergente, Dependente, Serialista e Reflexividade, indicando que os estilos cognitivos variaram em função da idade do estudante. Notou-se também que todas as relações entre as variáveis estilos cognitivos e desempenho na prova de matemática estão abaixo de 0,3, indicando correlações fracas, não sendo possível afirmar a existência de relações entre elas. Observou-se ainda a existência de relações entre o desempenho dos estudantes na prova de matemática e as estratégias de solução de problemas, indicando que a ?estratégia para frente", a mais empregada pelos sujeitos, mostrou-se adequada para a solução dos problemas propostos pelo estudo. Com relação à existência ou não de possíveis relações entre os estilos cognitivos e as estratégias de solução de problemas, foi verificada uma única relação encontrada entre o estilo Reflexibilidade e as estratégias de solução de problemas empregadas no problema A, na amostra masculina. Em relação ao gênero feminino observa-se 6 correlações de pequena intensidade (r < 0,3), e apesar das relações apresentarem pouca intensidade, pode-se ponderar que existe uma tendência do gênero feminino em apresentar relações entre as variáveis estilo e estratégias. Apesar de tais resultados, torna-se necessária uma análise mais aprofundada em relação aos estilos cognitivos e as estratégias utilizadas, na tentativa de explicar as causas das relações encontradas, mesmo que baixas. Por esses dados encontrados, enfatiza-se a importância de replicar este estudo em amostras maiores, para a confirmação ou não dos resultados obtidos. / Abstract: This study aimed to verify the existence of relationships between cognitive styles and strategies of problem solving, cognitive styles and the variables gender and grade, cognitive styles and student performance in problem solving, as well the performance and type of strategy used in solving problems. To this end, was asked 168 students to the first, second and third grade of High School from two public schools that responded to a questionnaire information, a Likert type scale to categorize the cognitive styles, and a mathematical test, addressing the concepts and procedural algebraic and arithmetic. Through analysis of cognitive styles in relation to this study, the predominance of divergence on the convergence of thought. And from the analysis of means of variables, cognitive style and gender, the means obtained indicated that women compared to men, had a higher prevalence in the styles Reflective and Serialist and lower in cognitive styles Impulsive and of Field-Independence. Both gender were predominant style Field-Dependent, Serialist, Reflective. There was still significant differences in cognitive styles, when considering the series of study revealed between the groups according to cognitive style Convergent, Dependent, Serialist and Reflective, indicating that the cognitive styles vary depending on the student's age. It was also noted that all relations between the variables cognitive styles and performance in mathematical test are below 0.3, indicating weak correlations, it is not possible to affirm the existence of relations between them. We also observed the existence of relationships between student performance on the mathematical test and strategies of problem solving, indicating that the "forward strategy", the most used by the subjects, proved adequate to solve the problems proposed the study. Regarding the existence of possible relationships between cognitive styles and strategies of problem solving, there was a unique relationship founded between the style Reflective and strategies of problem solving used in problem A, the male sample. Regarding females there were 6 low intensity correlations (r <0.3), and although relations were low intensity, can be considered that there is a tendency for females to present relationships between variables style and strategies. Despite these results, it is necessary a further analysis in relation to cognitive styles and strategies used in an attempt to explain the causes of the relationships found, even that low. For these findings, we emphasize the importance of replicating this study in larger samples to confirm or not the results. / Doutorado / Psicologia, Desenvolvimento Humano e Educação / Doutor em Educação