Parametric equations are used to represent the pathway of an object in terms of time or another changing variable. This allows, for example, for equations that are written using two variables to be examined in terms of the passage of time. In this paper the author examines two traditional application problems whose solutions can be enriched through the use of parametric equations. In the first, the falling ladder problem, a ladder is leaned against a wall then pulled away with a constant velocity. Deriving parametric equations for this scenario permits the pathway of the ladder to be plotted. Parametric equations also make it possible for the horizontal and vertical velocities of the ladder to be examined separately. The second problem is that of maximizing the length of a ladder that can fit around a hallway corner. In this problem an envelope algorithm is first developed, then parametrized to further investigate this scenario. Using these two situations, this report ultimately shows how parametric equations can be used to give a more thorough approach to some of today’s most classic calculus problems. / text

Research has shown that students interest in academics declines significantly with age, especially in the areas of math and science (Kahle et al., 1993; Wigfield, 1994; Wigfield and Eccles, 1992). One approach to combating this problem is by using new technologies to engage students who otherwise would not be interested in learning. In the AquaMOOSE project, 3D graphical technology is combined with a constructionist learning philosophy to create an environment where students can creatively explore new mathematical concepts. The AquaMOOSE socio-technical system has been developed using an iterative design process. Three formal studies were conducted to assess the effectiveness of the system, as well as several smaller scale evaluations. The first study was conducted during a six-week summer program where students were able to use the AquaMOOSE system during their free time. The second study explored different learning issues in the context of a comparison-class study at a local high school where one section learned about polar coordinates using standard curriculum materials and an equivalent section learned the same material using a curriculum designed specifically around the AquaMOOSE system. The final study of the AquaMOOSE system was in an eight-week after-school program at a local high school where a balance between structure and creative freedom was explored. In this thesis, the iterative design and evaluation of the AquaMOOSE socio-technical system is presented. Evidence from this process is used to suggest implications of using 3D technology and constructionist philosophy for teaching complex mathematical content. The findings presented address issues of using constructionist learning environments for complex content and the tradeoffs of using 3D technology for educational systems.

Made available in DSpace on 2016-04-27T16:57:45Z (GMT). No. of bitstreams: 1 EDM - Carlos Roberto da Silva.pdf: 4174831 bytes, checksum: 1d38259ebfb2955961be336c2a3a1d23 (MD5) Previous issue date: 2006-10-16 / Secretaria da Educação do Estado de São Paulo / This research has as objective verify if in a computer science environment allows to the student to recognize some curves properties through representations and graphical interpretations in dynamic way with the use of parameters for one comprehension better of its equations. We verified that the articulating between the points of view cartesian and parametric and the conversions among some registers of semiotic representation it makes the student think about the correlation that exists between some properties geometric of the plane curves and its cartesian or parametric equations. For this research we elaborate a didactic sequence based on some topics of the Didactic Engineering and we apply during five sessions in a group of 10 students taking the third year of high school. We verified that the graphic constructions of some plane curves varying the real values of its parameters in its equations for the development of an GIF (Graphic Information Format), they allow the students to observe the geometric effect caused by this variation what it favors the understanding of the parameter notion in analytical geometry / Esta dissertação tem por objetivo verificar se um ambiente informático permite ao aluno reconhecer algumas propriedades de curvas, por meio de representações e interpretações gráficas de maneira dinâmica, com o uso de parâmetros, para uma melhor compreensão de suas equações. Identificamos que a articulação entre os pontos de vista cartesiano e paramétrico e as conversões entre alguns registros de representação semiótica possibilitam ao aluno refletir sobre a correlação entre algumas propriedades geométricas de curvas planas e suas equações cartesianas ou paramétricas. Para esta pesquisa, elaboramos uma seqüência didática com base em alguns elementos de uma Engenharia Didática e aplicamos durante cinco sessões a um grupo de 10 alunos da 3ª série do Ensino Médio. Verificamos que as construções gráficas de algumas curvas planas, variando os valores reais de parâmetros em suas equações, para o desenvolvimento de um GIF animado, permitem ao aluno observarem os efeitos geométricos provocados pela sua variação, favorecendo o entendimento da noção de parâmetro na geometria analítica