Dissertations / Theses on the topic 'Strategies to solve geometric problems'

This paper presents the results of an investigation that aimed to analyze the strategies adopted by Aracajunian students of final year of elementary school to solving geometric problems. For this, we turn to problems taken from the books of |The Conquest Collection of Mathematics| authored by Giovanni Jr and Castrucci (2009) to develop an instrument that was initially applied to the students of 7th to 9th grades four municipal schools. After an examination of the responses submitted, a script was prepared to conduct semi-structured interviews with individuals who had different strategies through the first instrument collection. The theoretical assumptions were taken primarily from Polya (1978) for the understanding of mathematical problem geometric, their typology and possible resolution procedures. According to the examination of Polya (1978), a geometric problem characterized by ordering the contents geometry to solve it. The types of mathematical problems, according to the author can be classified from the utterance as routine, practical, and puzzle heuristic, and also for its solution are forms of determination and demonstration. Strategies to solve geometric problems highlighted in the book |The Art of Problem Solving| are using notation and formulas, as well as idealization or making figures. The results indicate that students have to geometrical problems responses, all three types by means of figures and then through arithmetic strategy. Records and algebraic strategies do not occur to students of Year 7, students are tentatively expressed by the following year and begin to gain prominence in the 9th grade classes. Students of years the different elementary school to solve routine problems similar to position geometry, in general, do not get the same success in the resolution, and the classes of 9th grade using guaranteed geometric strategy, while classes of Year 7, even if they have auxiliary notations demonstrate not feel secure about your solution, because their calculations up to justify their answers. Practical issues, applied to students in Year 7, related to the area have been resolved through the notion of perimeter, since the 8th grade students had good understanding of the concepts related to angles. In both cases there is a strong presence of geometric and arithmetic strategies. In short the figures are an important resource for these students develop their strategies with greater freedom of exposition, because through them, takes the stimulus to creativity and exercise for the establishment of solution plans.
O presente trabalho apresenta os resultados de uma investigação que teve como objetivo analisar as estratégias adotadas pelos alunos aracajuanos dos anos finais do ensino fundamental para resolução de problemas geométricos. Para isso, recorremos à problemas retirados dos livros da Coleção A Conquista da Matemática de autoria de Giovanni Jr e Castrucci (2009) para elaborar o instrumento que foi aplicado inicialmente aos alunos de 7º ao 9º anos de quatro escolas municipais. Após um exame das respostas apresentadas, foi elaborado um roteiro para realizar entrevistas semiestruturadas com os sujeitos que apresentaram estratégias diferenciadas por meio do primeiro instrumento de coleta. Os pressupostos teóricos foram tomados basicamente de Polya (1978) para o entendimento sobre problema matemático geométrico, sua tipologia e os possíveis procedimentos de resolução. De acordo com o exame de Polya (1978), um problema geométrico caracteriza-se por requisitar conteúdo da Geometria para resolvê-lo. Os tipos de problemas matemáticos, de acordo com o referido autor podem ser classificados a partir do enunciado como rotineiro, prático, enigma e heurístico, e também pela sua solução que são das formas determinação e demonstração. As estratégias para resolver problemas geométricos evidenciadas na obra A Arte de Resolver Problemas são uso de notação e de fórmulas, como também idealização ou confecção de figuras. Os resultados da pesquisa indicam que os alunos apresentam respostas aos problemas geométricos, de todos os três tipos, por meio de figuras e em seguida por meio de estratégia aritmética. Os registros e estratégias algébricas não ocorrem aos alunos de 7º ano, se expressam timidamente pelos alunos do ano sucessivo e começam a ganhar destaque nas turmas de 9º ano. Alunos de diferentes anos do ensino fundamental ao resolverem problema rotineiro similar sobre geometria de posição, em geral, não obtêm o mesmo sucesso na resolução, sendo que as turmas de 9º ano utilizam com garantia a estratégia geométrica, enquanto as turmas do 7º ano, ainda que disponham de notações auxiliares, demonstram não se sentir seguros sobre sua solução, pois apresentam até cálculos para justificar suas respostas. Os problemas práticos, aplicados a alunos de 7º ano, relacionados a área foram solucionados através da noção de perímetro, já os alunos de 8º ano, apresentam boa compreensão dos conceitos relacionados a ângulos. Em ambos os casos há forte presença de estratégias aritméticas e geométricas. Em suma as figuras constituem um importante recurso para esses alunos desenvolverem suas estratégias com maior liberdade de exposição, pois através delas, se dá o estímulo para a criatividade e o exercício para o estabelecimento de planos de solução.

Bibliography: leaves 104-108.
This study investigates how grade 12 students understand and solve geometric problems. A review of the literature on "how students learn and understand geometry" is used to develop a conceptual framework for discussion. The framework is used to assess students' level of understanding and to analyse their difficulties in solving geometric problems. The study was conducted at four low achieving schools in mathematics (based on student performance in the South African Senior Certificate Examination). It involved 267 students across the schools. The students' level of understanding was assessed through the use of two tests. These tests were designed to cover 80% of the grade 11 syllabus and involved the testing of a terminology framework (test 1) and problem solving exercises (test 2). Test 1 included 10 items were students were asked to complete statements, as well as 9 items where students were asked to write down properties from given sketches. Test 2 included 16 items of true or false responses.

La sélection de valeurs adéquates pour les paramètres des algorithmes bio-inspirés est un processus crucial pour obtenir de bonnes performances. C’est un processus complexe étant donné que les valeurs des paramètres dépendent en général du problème que l’on cherche à résoudre. Il est de plus très coûteux en temps, et les paramètres ne sont pas complètement indépendants entre eux. La première contribution de cette thèse consiste en une étude comparative des caractéristiques et performances des différentes méthodes connues dans la littérature pour ajuster les paramètres, en analysant leurs avantages et leurs inconvénients. Le plus grand désavantage de ces méthodes est qu’elles sont très coûteuses en temps. Cependant, elles permettent d’obtenir de bonnes estimations pour les paramètres, en concentrant la recherche dans des régions statistiquement prometteuses. Nous présentons de plus une analyse du potentiel de ces techniques en tant qu’aide dans le processus de conception d’algorithmes évolutifs. Même si ces techniques aident à prendre des décisions quant à la conception de l’algorithme, elles ne sont pas capables d’éliminer de l’algorithme les éléments qui ne contribuent pas à son bon fonctionnement. La contribution principale de la thèse se situe sur la capacité de déterminer les valeurs des paramètres à la volée. Nous proposons deux nouvelles méthodes pour améliorer les performances des algorithmes évolutifs : Ac et Sac. Ac réalise un contrôle de la population dans lequel un opérateur reçoit une récompense en fonction de l’amélioration de performance issue de son application. Sac est quant à lui orienté à un contrôle individuel dans lequel le taux d’un opérateur qui correspond à un individu dépend de l’efficacité qu’a eue cet opérateur dans l’amélioration des générations précédentes. Ces deux méthodes peuvent être incorporées sans introduire un coût significatif, et sans changement important dans la conception d’origine de l’algorithme. Les décisions prises durant la recherche se basent ainsi sur les informations disponibles sur le comportement passé, à partir d’un processus permettant à l’algorithme d’effectuer des changements de valeurs de paramètres qu’il juge nécessaires. L’application de ces techniques a permis à l’algorithme d’éliminer des composants non nécessaires à sa recherche, grâce à une utilisation minimale de ceux-ci. Nous proposons également la stratégie (C,n)-strategy connue pour réaliser un contrôle dans l’algorithme immunitaire CLONAG, permettant de réduire en bonne partie le nombre de paramètres sur lesquels repose son exécution, ainsi que d’améliorer son comportement grâce à la modification à la volée du nombre de cellules sélectionnées pour le clonage, ainsi que le nombre de clones produits. Les techniques appliquées à l’algorithme évolutif, comme celle appliquée à l’algorithme immunitaire cherchent un équilibre entre l’exploration et l’exploitation de l’espace de recherche en utilisant les paramètres sélectionnés pour réaliser le contrôle. A partir du travail réalisé dans cette thèse, nous pouvons conclure que les techniques d’ajustement des paramètres permettent effectivement de trouver un ensemble de valeurs des paramètres qui offre une bonne performance de l’algorithme de recherche. Ces techniques demandent une analyse antérieure de la définition de l’espace de recherche pour les valeurs des paramètres, et requièrent de plus un temps d‘exécution considérable. Néanmoins, à partir des informations données par ces techniques d’ajustement, il est possible de prendre des décisions concernant la conception de l’algorithme que l’on veut ajuster. En ce qui concerne les techniques de contrôle de paramètres, nous pouvons conclure qu’elles permettent d’améliorer l’efficacité de l’algorithme et de réduire l’effort d’ajustement des paramètres car elles réduisent le nombre de paramètres à ajuster, qu’elles sont capables d’ajuster automatiquement les paramètres de l’algorithme sans modifier la conception d’origine de l’algorithme, et enfin, qu’elles ajoutent une quantité minime de paramètres additionnels à l’algorithme. Le développement de techniques capables de travailler avec n’importe quel type de paramètres demandant une définition de l’information d’entrée et qui soient capables de stopper à temps la recherche afin de ne pas perdre de ressources, reste un terme de recherche ouvert. Nous avons montré dans cette thèse qu’il est possible de réaliser un contrôle à la volée des paramètres des algorithmes basés sur des populations. Une perspective de recherche intéressante consiste à arriver à automatiser l’identification du rôle des composants de l’algorithme afin de pouvoir concevoir automatiquement les stratégies de contrôle
Selection of proper values for parameters of bio-inspired algorithms is crucial to get good performance. To find these values is a complex process because parameter values depend on the problem that is being solved. Moreover, it is a high time consuming task and parameters are interrelated values. The first contribution of this thesis is the comparative study of features and performance of some automated parameter tuning methods. Advantages and disadvantages of these methods were identified. The main disadvantage of tuning methods is that they are high consuming time process. These methods search for parameters values focusing on statistically promising areas of the parameters search space. We also present an analysis of the potential of using these techniques to support the design process of evolutionary algorithms. In most cases, tuning methods are not able to discard elements that do not contribute to their operation. The main contribution of this thesis is related to the ability of varying parameter values during the execution of the algorithm. These kinds of approaches are known as parameter control methods. Two methods are proposed in this thesis for improving the performance of evolutionary algorithms by controlling its operators rates : Ac and Sac. Ac works in a population level rewarding operators that perfom good transformations during the iteration and punishing the others. Sac works in an individual level rewarding or punishing operators according to the performance of their applications in each individual. These both control methods can be incorporated to the evolutionary algorithm without introducing a meaningful cost in terms of time and without changing the original structure of the evolutionary algorithm. Variations of parameter values are based on current and past information about the operators performance. The incorporation of these techniques allows the algorithm to decide when and how much use each operator during the search process. Moreover, we propose the (C,n)-strategy for the CLONALG artificial immune system. In this case we control the amount of selected cells and clones produced in each iteration. The application of our control method allows us to reduce the amount of parameters to tune for the algorithm. Our control technique is able to manage the relationship between intensification and diversification stages of the search by varying the values of these parameters. From this work, we can conclude that parameter tuning techniques are able to find parameter values that show a good performance on the tuned algorithm. Most of these techniques require a previous analysis of the parameter search spaces and, in general, they require a considerable amount time. However, they can be able to make decisions about the design of the evolutionary algorithm. About the incorporation of parameter control strategies, we can conclude that it is possible to improve the efficiency of the algorithm, reducing the tuning effort by decreasing the number of parameters of tune. It is possible to vary parameter values during the search without changing the original design of the algorithm. Finally, the proposed techniques add a minimum amount of easily tunable new parameters. The development of tuning techniques able to work with any kind of parameters and capable to determine proper stopping criteria in order to do not waste resources is still an open research area. In this thesis we have shown that it is possible to perform on-line parameter variation in population-based algorithms. One interesting point of research is the automatic identification of the role of different components of the algorithm in order to automatize the design process of parameter control strategies

This action research study examined the influence mathematical strategies had on middle school students' mathematical ability. The purpose of this action research study was to observe students mathematical abilities and to investigate whether teaching students problem-solving strategies in mathematics will enhance student's mathematical thinking and their ability to comprehend and solve word problems. The study took place in an urban school in Orlando, Florida in the fall of 2004. The subjects will be 12 eighth grade students assigned to my intensive math class. Quantitative data was collected. Students' took a pre and post test designed to measure and give students practice on mathematical skills. Students worked individually on practice problems, answered questions daily in their problem solving notebook and mathematics journals. Results showed the effectiveness of the use of direct instruction and problem-solving strategies on at-risk students. 
M.Ed.
Other
Graduate Studies;
K-8 Math and Science MEd

碩士
中原大學
應用數學研究所
104
“Proof question” is the most difficult topic for teachers to teach in high school mathematics classes; and this is proved the most insufficient training for students nowadays. To compare to other topics, "proof question” is less interesting; however, students can be trained effectively to think logically and to inspire their observation skills and imagination by solving proof questions. The topic of this study is based on the plane geometry of ninth grade in junior high schools and takes the proof of plane geometry into consideration. The topic of this study: What are the strategies for solving proof questions of linear classical geometric? According to the topic of this study, there are several ways to solve the question are: “extensive midline, graphics features, auxiliary circle, graphic cut edges and corners.” With these strategies, the proof questions can be solved step by step. A decent knowledge is a prerequisite for solving any proof questions; in addition, proof questions usually have various solutions. The more solutions can be thought of when students have more experience. Importantly, teachers are also required to observe students’ ideas and reaction and give appropriate hints for students to think and learn. By having a broader vision given by the teachers, students’ way of thinking and development will be limited less. Overall, these strategies and techniques are specifically for straight geometric proofs, not applicable for any other kinds of questions. With these strategies skills, students can reach the objective of the requirement of nine-year integrated mathematics courses on the topic of Geometric: To understand the significance of proof.

碩士
中原大學
應用數學研究所
105
Geometry is always the most difficult problem which junior high school students encounter.Students even feel frustrated and may give up learning especially when facing the chapter of circle. As excellent educators, what we need to do is not only to observe the learning situation of the students but also to build up their ability of drawing and thinking about geometry, leading them how to deal with problems. The topic of the research is based on the Unit Two “Circle” of the ninth grader of junior high school. It aims to how to clarify the related questions, how to analyze the problem-solving strategies and how to instruct the students to figure out the problems. The purpose of the study is as follows: (a) The students know how to figure out the problems step by step and estabish the ability to write out the solutions of the proof questions. (b) The students have the ability to apply the geometric thinking to the related ideas about circle. (c) The students have the ability to extend the prove by using the result of the prove. (d) The students can understand the related properties of congruent circles and concyclic. The study only focuses on how to provide the problem-solving strategies by using the proof of congruent circles and concyclic. In teaching, it’s important to build up the prerequisite knowledge, guide the students to think by using the way of the proof of the fragment and build up the problem- solving strategies. So that we can achieve the inferential capability and the linking ability of the core competencies of the junior high school mathematics curriculum guidelines.

碩士
中原大學
應用數學研究所
105
Proportionality, which plays a very important role in geometric proof, contains plane geometry, parallelogram, trapezoid, similar figure, circle, and circumcenter, incenter, and centroid of a triangle in geometric range of junior high mathematics. The purpose of the study is to help junior high school students to solve the problems when dealing with the geometric proof questions relating to proportionality. In addition, it shows how teachers guide students, lead students to deal with the problems progressively by reviewing the concepts about geometry that they’ve learned in junior high school, comprehending the meaning of the questions, and using the clues from the text and the related experience learned before to solve the problems. Finally, students will know how to integrate what they’ve learned and applied the skills to other similar problems. Since the range of the proportionality is wide, the study mainly focuses on the classifications of parallel property, proportional segment property, and similarity. Other extend graphics are the secondary. And the relevant property of circle won’t be discussed here. Keywords：geometric proportion , parallel , proportional segment , similarity

The single-row facility layout problem (SRFLP) is concerned with finding the optimal linear placement of n departments with different lengths in a straight line. It is typically achieved by minimizing the cost associated with the interactions between the departments. The semidefinite programming (SDP) relaxation model that incorporates cutting planes proposed recently by Anjos, Kennings, and Vannelli (AKV) was considered a breakthrough in the field. This thesis presents a new SDP model AKV' and compares the two relaxations. The AKV' is largely based on the previous model, but it reduces the number of linear constraints from O(n³) to O(n²). Therefore, it reduces the computing time at the expense of a slightly weaker lower bound. However, AKV' is observed to pay off as the instance size increases. By examining the gap for both the AKV and AKV' relaxations, we notice that both relaxations generate very small gaps at the root node, which demonstrates the effectiveness of the relaxations. Six different strategies are presented to separate the cutting planes for the medium-sized SRFLP. In combination with the two SDP relaxations, we compare the six strategies using three instances of different characteristics. An overall best strategy is deduced from the computational results, but the best choice of relaxations and the best number of cuts added at each iteration changes depending on the characteristics of the instances. Two new cutting plane strategies are proposed for large instances. This allows the solution to optimality of new instances with 36 departments, which is higher than previously published results in literature. We also briefly point out how the computing time can vary greatly between different sets of data of the same size due to the characteristics of the department lengths.

碩士
樹德科技大學
金融系碩士班
103
The fertility rate in Taiwan declines over the years so how to deal with the influences of education industry from low birth rate is important. Having an English cram school is easy because the investment doesn’t need too much. However, operating well is difficult due to the industry competing closing. The key to get rid of the difficulty of low birth rate and operate well is to keep the stable benefits. The study will analyze the development and current situation of English cram schools in Taiwan and J Cram School is as a case study. The qualitative research is implemented to inquiry business operating model and the interviewees are two managers and teachers in respectively branch of J Cram School in Kaohsiung. On the other hand, the questionnaires are done by a sampling of fifty parents. Then the result shows that the strategies of J Cram School are three parts, hardware, software, and human resources. Also, the suggestions, based on the research result, are offered for further operating, including creating the interactive website (e-learning), teaching videos online for making up the missed lessons, increasing the teaching hours of the foreign teachers, having a trip to learn English abroad, designing the e-books for student version, assigning more oral homework, using value strategy and dealing with administrative matters flexibly.

碩士
國立臺北教育大學
特殊教育學系碩士班
104
Abstract This study mainly focuses on investigating the influence of using line-diagram strategy in combination with self-instruction strategy on the learning achievements in math-word problem of elementary school students of a resource class. This study method is primarily based on action research and three elementary school students among resource classroom are selected as the subjects. The Kang-Hsuan 3th-grade math teaching materials are re-edited and the dual-cycle teaching by taking advantage of line-diagram in combination with self-instruction strategy is used. This researcher summarized and analyzed the outcomes by instruction, observation, evaluation, and reflection. The research results are concluded as follows: 1.Line-diagram in combination with self-instruction strategies can enhance the learning efficiency in math-word problems of students among resource class. 2.Line-diagram in combination with self-instruction strategies can improve the learning motivation in math-word problems of students among resource class. 3.The action research on line-diagram in combination with self-instruction strategies can expand research’s pedagogical knowledge. According to the research results, the researcher has made some corresponding suggestions for future instructional activities as the research and teaching references