This thesis is the report of an experiment carried on under the direction of Harry Huffman, Professor of Business Education at Virginia Polytechnic Institute, using material in Programmed Business Mathematics which was written by Dr. Huffman especially for post high school students and college freshmen. The programmed material used in the experiment consisted of the first 239 frames covering the basic fundamentals of addition, subtraction, multiplication, and division. The main emphasis of the program is on building speed and accuracy in applying the skills and concepts related to these basic processes. The subjects of the experiment were eleven high school students from grades 9, 10, and 12, at Eastern Mennonite High School. The purpose of the experiment was to obtain these data: 1. What kind of errors do high school students make on the program? 2. What changes need to be made in the program to adapt it for use on the high school level? 3. Can the program be adjusted so that high school students can work through it with five percent or less of errors? The writer found that errors were grouped under these major headings and in this order: Lack of comprehension of concepts introduced, carelessness, and computational (errors of addition, subtraction, multiplication, and division). The frames were revised three times during the experiment. Revision consisted of adding prompts of underlining or new words, rewriting frames, or constructing new frames. Two students went through the original frames: average errors, 6.56 percent. Three students worked with the first revision: average errors, 11.58 percent. Four students worked with the second revision: average errors, 3.13 percent. Two students worked with the third revision: average errors, 2.56. Although the error rate did not drop consistently, it did drop from a high of 11.58 percent on the second round of testing to 2.56 percent for the final round, a decrease of 9.02 in percent. Individual differences in I.Q., previous training, and the manner in which the student-editors worked undoubtedly account for part of the variation in error percentages. The results of the experiment show that it is possible for high school students to work on the fundamentals of business mathematics frames with an error limit of five percent or less. / Master of Science

An experiment to adjust a programmed college business mathematics text to the level of comprehension suitable for high school students of low ability was prompted by the writer’s awareness of the need for a similar text for high school business majors. The writer was especially concerned with those students who lack ability to attend college, but who go into business offices seeking employment-- positions which invariably require a knowledge of the fundamentals of business mathematics. The case study approach was used in conducting the experiment. The research involved a programmed text of 432 frames, entitled Fundamentals of College Business Mathematics, by Dr. Harry Huffman. Two major objectives were established as a basis for this study. The first objective was to adjust Huffman’s programmed business mathematics text to a level of comprehension of students of below average ability, so that their completed work, without a teacher's help, would be approximately 95 percent accurate. The second objective of the study was to determine whether students could retain the material covered in the programmed unit. This objective was achieved by means of intermediate tests given to each student throughout the unit. Four revisions of Huffman’s original programmed text were necessary to accomplish the objectives of the study. Five panels of student-editors, each panel comprised of three students of below average ability, participated in the research. Revisions to the original programmed text were made on the basis of incorrect responses by individual students working through the unit. Written and oral comments by these students were also taken into consideration in making revisions. Analysis of the data revealed the following results of the total experiment. A total of 104 new frames were added to the original program of 432 frames. This represented a 24.1 per cent increase of frames in the total unit which required an additional 208 responses. There was a total decrease in errors from Panel One to Panel Five of 5.20 per cent. The average score on the six intermediate tests increased 9.1 points from the beginning to the end of the experiment. Results of the experiment lead the writer to conclude that: 1. The error rate of successive panels dropped from 8.03 to 2.83 per cent. This represents a total decrease of 5.20 per cent from the beginning to the end of the experiment. Therefore, it was concluded that senior students, in high school, of below average ability can complete the adjusted programmed unit with an average error rate of less than 5 per cent. 2. Test score average for the panels rose from 78.2 to 87.3, an increase of 9.1 points, from the beginning to the end of the experiment. Therefore, it was concluded that students of below average ability can successfully perform on the retention tests used to measure the result of the programmed unit on fundamentals of business mathematics. / Master of Science

Submitted by Filipe dos Santos (fsantos@pucsp.br) on 2018-12-11T11:58:42Z No. of bitstreams: 1 Ricardo Antonio de Souza.pdf: 1542597 bytes, checksum: f061f50caaad575d2edaa18c72092440 (MD5) / Made available in DSpace on 2018-12-11T11:58:42Z (GMT). No. of bitstreams: 1 Ricardo Antonio de Souza.pdf: 1542597 bytes, checksum: f061f50caaad575d2edaa18c72092440 (MD5) Previous issue date: 2018-09-28 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / In recent years, we have noticed a large increase in the number of families in Brazil, thinking about how to contribute to the improvement of this situation, we decided to develop this research, which seeks to verify if financial mathematics, when developed by mathematical modeling as a teaching and learning strategy, can contribute to the development of students' financial behavior in a higher technology course in logistics and computer science. In the present research, we hypothesize that when financial mathematics is developed using this strategy, there is a positive influence on the understanding of the content worked and, consequently, influences the financial behavior of the subject. In order to validate our hypothesis, we use the modeling processes, which brings the student closer to the real scientific activity, that is, the student becomes a researcher, testing conjectures, formulating hypotheses, proving, constructing models, concepts, theories and socializing the results. According to this strategy, it is up to the teacher to provide favorable situations, so that the student in this effective action on knowledge, transform it into knowledge in this process of scientific search. Recent research shows that students who have already completed the discipline of financial mathematics in an undergraduate course usually do not use or associate the studied content with their financial life, so we try to answer the following research question: The use of mathematical modeling as a strategy for teaching and learning Financial Mathematics, could contribute to the development of Financial Education ?. As well as the following specific questions: What evidence of financial literacy can be identified after a training on Financial Mathematics centered on the modeling approach? What contributions of mathematical modeling can be identified for the construction of financial literacy? To answer them, we created a questionnaire to be applied to students of two technology courses of a public institution of higher education, located in the great ABC - SP. Being the first, technology in logistics and the second in computing. The students of the first course answered the questionnaire after attending a financial mathematics workshop, structured by mathematical modeling. In order to have a comparative parameter of answers, we apply the same questionnaire to students of the second course, but these without participating in the workshop of financial mathematics. With the answers given to the questionnaire, we performed a cohesive analysis with the help of the CHIC software, and identified traits of development of financial education by our target audience / Nos últimos anos, notamos um grande aumento do individamento das famílias brasileiras, pensando em como contrubuir para a melhora desse quadro, decidimos desenvolver esta pesquisa, que busca verificar se a matemática financeira, quando desenvolvida por modelagem matemática como estratégia de ensino e aprendizagem, pode contribuir para o desenvolvimento do comportamento financeiro dos alunos de um curso superior de tecnologia em logística e informática. Na presente pesquisa, temos por hipótese que, quando a matemática financeira é desenvolvida utilizando esta estratégia, existe influência positiva na compreensão dos conteúdos trabalhados e, por consequência, influencia o comportamento financeiro do sujeito. Buscando validar nossa hipótese, utilizamos os processos de modelagem, a qual aproxima o aluno da atividade científica verdadeira, ou seja, o aluno se torna um pesquisador, testando conjecturas, formulando hipóteses, provando, construindo modelos, conceitos, teorias e socializando os resultados. Segundo essa estratégia, cabe ao professor, providenciar situações favoráveis, de modo que o aluno nessa ação efetiva sobre o saber, o transforme em conhecimento nesse processo de busca científica. Pesquisas recentes nos mostram que alunos que já cursaram a disciplina de matemática financeira em um curso de graduação, normalmente não utilizam ou associam o conteúdo estudado com a sua vida financeira, assim, procuramos responder a seguinte questão de pesquisa: A utilização da modelagem matemática como estratégia de ensino e aprendizagem de Matemática Financeira, poderá contribuir para o desenvolvimento da Educação Financeira?. Bem como as seguintes questões específicas: Que indícios de letramento financeiro podem ser identificados após uma formação sobre Matemática Financeira centrada na abordagem pela modelagem? Que contribuições da modelagem matemática podem ser identificadas para a construção do letramento financeiro? Para respondê-las, montamos um questionário a ser aplicado para alunos de dois cursos de tecnologia de uma instituição pública de ensino superior, localizada no grande ABC – SP. Sendo o primeiro, tecnologia em logística e o segundo em informática. Os alunos do primeiro curso responderam o questionário após participarem de uma oficina de matemática financeira, estruturada pela modelagem matemática. Para termos um parâmentro comparativo de respostas, aplicamos o mesmo questionário para alunos do segundo curso, mas estes sem participarem da oficina de matemática financeira. Com as respostas dadas ao questionário, realizamos uma análise coesitiva com o auxílio do software CHIC, e identificamos traços de desenvolvimento de educação financeira por parte do nosso público alvo

Thesis (MComm (Mathematics))--University of Stellenbosch, 2007. / The word numéraire refers to the unit of measurement used to value a portfolio of assets. The change of numéraire technique involves converting from one measurement to another. The foreign exchange markets are natural settings for interpreting this technique (but are by no means the only examples). This dissertation includes elementary facts about the change of numeraire technique. It also discusses the mathematical soundness of the technique in the abstract setting of Delbaen and Schachermayer’s Mathematics of Arbitrage. The technique is then applied to financial pricing problems. The right choice of numéraire could be an elegant approach to solving a pricing problem or could simplify computation and modelling.

CAPES / Este trabalho apresenta uma proposta diferenciada de abordagem da matemática financeira desenvolvida no Ensino Médio, com foco na Educação Financeira. Busca-se através da inserção de textos, da análise de alguns produtos financeiros e da interpretação de problemas contribuir para a formação financeira dos alunos. Objetivando tornar a Matemática Financeira mais acessível para o dia a dia, s˜ao disponibilizadas e estudadas algumas ferramentas de uso pratico, como a calculadora do cidadão, disponível para computadores e celulares. Assim como planilhas eletrônicas e o Software Geogebra, que permitem comparar e analisar os custos financeiros de consórcios, financiamentos e aplicações financeiras, além de serem aliadas na elaboração e controle do orçamento pessoal e familiar. Ao aplicarmos a sequência de ensino proposta em uma turma do ensino médio do período noturno, detectamos algumas dificuldades que limitaram a aprendizagem dos conceitos de Matemática Financeira. Contudo identificamos progresso com relação a Educação Financeira dos alunos, observações estas feitas por meio de análises qualitativa e quantitativa a partir das anotações no diário de campo e também de respostas dadas pelos alunos em formulário específico. / This paper presents a differential approach of financial mathematics developed in high school, focusing on financial education. We seek through the insertion of texts, analysis of some financial products and by the interpretation of problems to contribute to the financial training of students. Aiming to make financial mathematics more attractive and accessible to the day-today, there are available for studies some practical tools like the citizen calculator which are available for computers and mobile phones. As well as spreadsheets and software Geogebra, that allow you to compare and analyze the financial costs of consortia, financing and financial applications,in addition to being allied in the preparation and control of personal and family budget. By applying the teaching sequence proposed in a nocturnal class of high school, we realized some difficulties that have limited the financial learning of math concepts.Yet identified progress regarding the financial education of students. These observations were made through qualitative and quantitative analysis from notes in the field diary and also answers given by students in specific form.

In this thesis we establish a relationship between the Potential Approach to interest rates and the Market Models. This relationship allows us to derive the dynamics of forward LIBOR rates and forward swap rates by modelling the state price density. It means that we are able to secure the arbitrage-free condition and positive interest rate feature when we model the volatility drifts of those dynamics. On the other hand, we develop the Potential Approach, particularly the Hughston-Rafailidis Chaotic Interest Rate Model. The early argument enables us to infer that the Chaos Models belong to the Stochastic Volatility Market Models. In particular, we propose One-variable Chaos Models with the application of exponential polynomials. This maintains the generality of the Chaos Models and performs well for yield curves comparing with the Nelson-Siegel Form and the Svensson Form. Moreover, we calibrate the One-variable Chaos Model to European Caplets and European Swaptions. We show that the One-variable Chaos Models can reproduce the humped shape of the term structure of caplet volatility and also the volatility smile/skew curve. The calibration errors are small compared with the Lognormal Forward LIBOR Model, the SABR Model, traditional Short Rate Models, and other models under the Potential Approach. After the calibration, we introduce some new interest rate models under the Potential Approach. In particular, we suggest a new framework where the volatility drifts can be indirectly modelled from the short rate via the state price density.

Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Humanities, School of Education, 2013 / Mathematics-for-teaching (MfT) is complex, multi-faceted and topic-specific. In this study, a Financial Mathematics course for pre-service secondary mathematics teachers provides a revelatory case for investigating MfT. The course was designed and taught by the author to a class of forty-two students at a university in South Africa. Eight students, forming a purposive sample, participated as members of two focus tutorial groups and took part in individual and group interviews. As an instance of insider research, the study makes use of a qualitative methodology that draws on a variety of data sources including lecture sessions and group tutorials, group and individual interviews, students’ journals, a test and a questionnaire. The thesis is structured in two parts. The first part explores revisiting of school mathematics with particular focus on compound interest and the related aspects of percentage change and exponential growth. Four cases are presented, in the form of analytic narrative vignettes which structure the analysis and provide insight into opportunities for learning MfT of compound interest. The evidence shows that opportunities may be provided to learn a range of aspects of MfT through revisiting school mathematics. The second part focuses on obstacles experienced by students in learning annuities, their time-related talk, as well as their use of mathematical resources such as timelines and spreadsheets. A range of obstacles are identified. Evidence shows that students use timelines in a range of non-standard ways but that this does not necessarily determine or reflect their success in solving annuities problems. Students’ use of spreadsheets reveals that spreadsheets are a powerful tool for working with annuities. A key finding with regard to teachers’ mathematical knowledge, and which cuts across both parts of the thesis, is the importance of being able to move between compressed and decompressed forms of mathematics. The study makes three key contributions. Firstly, a framework for MfT is proposed, building on existing frameworks in the literature. This framework is used as a conceptual tool to frame the study, and as an analytic tool to explore opportunities to learn MfT as well as the obstacles experienced by. A second contribution is the theoretical and empirical elaboration of the notion of revisiting. Thirdly, a range of theoretical constructs related to teaching and learning introductory financial mathematics are introduced. These include separate reference landscapes for the concepts of compound interest and annuities

Este trabalho apresenta um estudo matemático detalhado dos sistemas de financiamento habitacional atualmente praticados no Brasil, estando alinhado com a Estratégia Nacional de Educação Financeira (ENEF), política de Estado de caráter permanente, instituída pelo Decreto Federal 7.397/2010, que tem como objetivo contribuir para o fortalecimento da cidadania através de ações que ajudem a população a tomar decisões financeiras mais autônomas e conscientes. A inserção da Educação Financeira no Ensino Fundamental e Médio, como conteúdo obrigatório da disciplina de Matemática, foi objeto de projeto de Lei que tramitou na Câmara e no Senado entre 2009 e 2013, tendo sido, por fim, rejeitado. Não obstante, centenas de escolas em todo o Brasil, públicas e particulares, já vêm ministrando o assunto, de forma obrigatória ou mesmo como tema extracurricular. Neste contexto, posto que a Matemática é uma ferramenta indispensável na compreensão de diversos problemas ligados a Educação Financeira, o presente trabalho também é voltado para o ensino, principalmente como fonte de consulta daqueles profissionais que atuam na divulgação da cultura da Educação Financeira no País. Ao contratar um financiamento habitacional, o cidadão precisa optar entre o Sistema de Amortização Constante (SAC) e o Sistema Francês de Amortização (Tabela Price). Porém, muitas vezes, não dispõe de informações suficientes para realizar a escolha mais adequada ao seu perfil financeiro. Este trabalho objetiva também preencher esta lacuna, apresentando ao cidadão orientações uteis para sua tomada de decisão, podendo ser visto, desta forma, como uma contribuição social. Isto se consegue, por um lado, através da obtenção de uma série de resultados Matemáticos para os dois sistemas de amortização vigentes, e, por outro, fornecendo respostas a d ́ vidas frequentes sobre o contrato de financiamento, tais como amortização do saldo devedor, alteração da data de vencimento e pagamento de prestações em atraso. / This work presents a detailed mathematical study of housing finance systems currently practiced in Brazil and is aligned with the National Strategy for Financial Education, a State policy established by Federal Decree 7.397/2010, which aims to contribute to the strengthening of citizenship through actions that help the population to take more autonomous financial decisions and aware. A law to include financial education in primary and secondary schools as an obligatory subject of mathematic, was analysed in the House of Representatives and Senate, between 2009 and 2013, and was finally rejected. Nevertheless, hundreds of schools throughout Brazil, public and private, are already teaching this subject, as a basic subject or even as an extracurricular subject. In this context, since mathematics is an indispensable tool in understanding various problems related to financial education, this work is also for teaching, mainly as a source of consultation of those professionals who work in spreading the culture of financial education in the country. By hiring a loan to buy a propriety, citizens need to choose between the Constant Amortization System and the French Amortization System (Price Table). However, often there are not sufficient informations to make the best choice to a specific financial profile. This work aims to also fill this gap, presenting to the citizen useful guidance for its decision making, can be seen, therefore, as a social contribution. This is achieved, on the one hand, by obtaining a series of theoretical mathematical results for the two existing amortization systems, and on the other, providing answers to frequently asked questions about the financing agreement, such as amortization of the debt, change the due date and payment of installments in arrears.

CAPES / Desenvolvemos um trabalho visando habilidades relacionadas à compreensão, investigação e a contextualização de situações problemas, caracterizando o pensar matemático na construção do conhecimento. Na medida em que abordamos situações matemáticas percebe-se o quanto é significativo ensinar e aprender. Tratamos a matemática financeira como um estudo de funções. Progressões, aritmética e geométrica, são tratadas como funções afim e exponencial, respectivamente, para um domínio discreto. Faz-se análise de gráficos e tabelas com o intuito de compreender e investigar situações de financiamentos. Através de uma pesquisa com a comunidade local orientamos os alunos quanto à importância de se ter uma educação financeira adequada para tomada de decisões. / We develop a job seeking skills related to the understanding, research and contextualization of problem situations, featuring mathematical thinking in knowledge construction. Insofar as situations mathematical approach realizes how teaching and learning is significant. We treat financial mathematics as a study of functions. Progressions, arithmetic and geometry are treated as, affine and exponential functions, respectively for discrete domain. It is analyzing graphs and tables in order to understand and investigate financing situations. Through a survey of the local community have the students about the importance of having a financial education for decision making.

CAPES / Este trabalho teve como objetivos investigar como é feita a abordagem da Matemática Financeira nos Livros Didáticos e a opinião dos Professores de Matemática do Núcleo Regional de Educação de Guarapuava (NRE) sobre o Ensino de Matemática Financeira e como está sendo realizada a abordagem nas escolas públicas da região. Como objetivo indireto o presente trabalho buscou propiciar um momento de reflexão junto aos Professores do NRE de como está o trabalho com Matemática Financeira em sala de aula. Na pesquisa que se apresenta, fizemos um estudo introdutório sobre o início das relações comerciais e financeiras em nossa humanidade. Analisamos a maneira como os livros didáticos disponibilizados através do Programa Nacional do Livro para o Ensino Médio (PNLEM) 2015 trazem o conteúdo de Matemática Financeira no contexto da Matemática escolar, assim como a opinião dos Professores com relação a esse conteúdo. Para isso, foram analisados os livros didáticos disponibilizados através do PN- LEM 2015 e realizada uma pesquisa quantitativa com Professores de Matemática que atuam na região de abrangência desse Núcleo Regional. Os resultados da pesquisa apontaram que os livros didáticos apresentam algumas deficiências em relação ao conteúdo de Matemática Financeira, mas evoluíram ao longo do tempo e os Professores consideram importante o Ensino de Matemática Financeira, porém poucas aulas ainda são destinadas para o trabalho com o tema. / This study aimed to investigate how is the financial mathematics approach in textbooks and the opinions of Teachers of the Regional Education Center (NRE) about Financial Mathematics Teaching and how it is being carried out the approach in public schools in the region. As an indirect aim of this study searched to provide a moment of reflection at the NRE’s Teachers of how it has been worked with Financial Mathematics in the classroom. In this research it was did an introductory study about the beginning of commercial and financial relations in our humanity. It was analyzed the way the textbooks available through the National Book Program for Secondary Education (PNLEM) 2015 bring the Financial Mathematics content in the context of school mathematics, as well as the teachers’ opinion about this content. For this, they were analyzed six textbooks available through the PNLEM 2015 and conducted a quantitative survey with Mathematics Teachers who work in the coverage area of this Regional Center. The search results showed that the textbooks have some failings regarding the Financial Mathematics content, but that have evolved over time and the teachers consider important the Teaching of Mathematics Financial, but few classes are also designed to work with the theme.