Um estudo sobre a aprendizagem de alguns conceitos algébricos e geométricos

The objective of the present work was develop a study about the writing and the algebraic manipulation of symbolical expressions for perimeter and area of some convex polygons, approaching the properties of the operations and equality, extending to the obtaining of the formulas of length and area of the circle, this one starting on the formula of the perimeter and area of the regular hexagon. To do so, a module with teaching activities was elaborated based on constructive teaching. The study consisted of a methodological intervention, done by the researcher, and had as subjects students of the 8th grade of the State School Desembargador Floriano Cavalcanti, located on the city of Natal, Rio Grande do Norte. The methodological intervention was done in three stages: applying of a initial diagnostic evaluation, developing of the teaching module, and applying of the final evaluation based on the Mathematics teaching using Constructivist references. The data collected in the evaluations was presented as descriptive statistics. The results of the final diagnostic evaluation were analyzed in the qualitative point of view, using the criteria established by Richard Skemp s second theory about the comprehension of mathematical concepts. The general results about the data from the evaluations and the applying of the teaching module showed a qualitative difference in the learning of the students who participated of the intervention

As geometrias urbana e isoperimétrica: uma alternativa de uso em sala de aula

The main goal of the present study is to propose a methodological approach to the teaching of Geometry and, in particular, to the construction of the concepts of circle (circumference) and ellipse by 7th and 8th grade students. In order to aid the students in the construction of these concepts, we developed a module based on mathematical modeling, and both Urban Geometry (Taxicab Geometry) and Isoperimetric Geometry. Our analysis was based on Jean Piaget's Equilibrium Theory. Emphasizing the use of intuition based on accumulated past experiences, the students were encouraged to come up with a hypothesis, try it out, discuss it with their peers, and derive conclusions. Although the graphs of circles and ellipses assume different shapes in Urban and Isoperimetric Geometry than they do in the standard Euclidian Geometry, their definitions are identical regardless of the metric used. Thus, by comparing the graphs produced in the different metrics, the students were able to consolidate their understanding of these concepts. The intervention took place in a series of small group activities. At the end of the study, the 53 seventh grade and the 55 eighth grade students had a better understanding of the concepts of circle and ellipse

Incorporação de lodo de esgoto na massa cerâmica para fabricação de tijolos maciços: uma alternativa para a disposição final de resíduo

The disposal of sewage sludge is a growing problem face up to management of sanitary sevices. Otherwise, because its making process characteristic, the Ceramic Industry can tolerate the presence of this wastes as raw material. This study has as object to confirm the use of the sewage sluge in the Ceramic Industry like a sustentable alternative for its disposal. Futhermore, this study quests to evaluate the maximum proportion for incorporation of sludge wich result in technically and enviromentally suitable bricks. For found this proportion, the research consisted of (1) making of bricks in full scale, adde up 0%, 5%,10%, 15%, 20%, 25%,30%, 35% e 40% sludge, with size 220x105x45 mm, hand-molded by rammer and baked by industrial kiln; and (2) tecnical and enviromental evaluation of this bricks, according to Brazilian norms. The raw material uses were two distinct clays come from Goianinha/RN and sewage comes from a septic system tank and pumped into tank vehicle, of Natal/RN. The technical evaluation allowed to conclude the addiotion of the sludge brings about signifcant lost of mass and the water absorption grew up according to increase of sludge: every sludge-amended clay bricks absorved more water than control group. Thus, the compressive strength was signicantly decreased because the increase of sludge: bricks with 5% sludge added lost 45% of strength achieved at control group; the bricks made with 10 and 20% lost almost 70% of bigger strength. With up to 25% sludge added to the bricks, the streght decreased over 90%. Concerning heavy metal leaching, the two maximum proportion wich have tecnical approval, it means bricks sludge added with 15 and 20%, can say there is no risk of enviromental contamination using those bricks. This way, in this work context, it can to conclude the maximum proportion atends the technical and enviromental criterion is 20%

Caracterização dos resíduos esgotados de sistemas de tratamento individual de esgotos de Natal

In Natal still dominates the use of individual disposal systems for domestic sewage, once only 29% of the city has a sewarage system. Wastes that are accumulated in these individual treatment systems should be exhausted periodically, service provided by collector entrepreneurs. Some of these companies causing major damage to the environment. In Natal, only two companies have their own septage (RESTI) treatment system, which were designed with parameters from domestic sewage generating strain and inefficient systems. Therefore, the characterization becomes essential as a source of parameters for their design. Thus, this work presents the physical-chemical and microbiological characterization of waste pumped from individual sewage treatment systems. Samples collections were made weekly from 5 different trucks at the reception point on the treatment plant on the point of the preliminary treatment. From each truck it was taken 5 samples during the discharge in order to make a composite sample. Afterwards, samples were carried out to laboratory and analyses for determination of temperature, pH, conductivity, BOD, COD, nitrogen (ammonia e organic), alkalinity, oils, phosphorus, solids, faecal coliforms and helminth egg. The results were treated as a single database, and ranked according to its generating source (multi and single house, lodging, health, services and / or food), area of origin (metropolitan, south and north) and type of system (cesspits, septic tank and / or sink). Through these data it was possible to verify that the type of system adopted by most in Natal and the metropolitan region is cesspit, besides to confirm the difference between the septage of areas with a population have different social and economical characteristics. It was found that the septage have higher concentrations than domestic sewage, except for thermotolerant coliforms that showed concentrations of 1,38E+07. Among the parameters studied, is the median values identified for COD (3,549 mg / L), BOD (973mg / L) and total solids (3.557mg / L). The volatile fraction constitutes about 70% of the total solids of the septage. For helminths has been a median of 7 eggs/L. In general, the characteristics of the waste followed the variability found in the literature reviewed for all variables, showing high amplitudes

Avaliação da eficiência de lagoas de estabilização no tratamento de resíduos esgotados de fossas sépticas

Because of disability in public policy development in mind to attend issues of sanitation in the municipalities, companies known as "clean-blue" appeared proposing to solve a simple collection and management of wastewater produced in single or multifamily residences, commercial, hospitals, etc. In the case of an activity in which there are no worries about the fate of sewage, emerged some doubts about the degree of health and environmental safety in these companies. Traditionally, most of them makes the provision of waste depleted soil or wetland, open, usually located on the outskirts of cities (MENESES, 2001). In turn, the sludge from septic tanks exhausted, provided no technical criteria - in the soil, rivers and as an agricultural fertilizer put in risk the health of the population and environmental quality. This work was entered in the search network 5 of the Notice of the Research Program in Sanitation - PROSAB-5, aimed to study the theme 'Characterization and study of alternative ways of treating sludge from septic tanks in the city of Natal, RN', proposing to evaluate the performance of the use of stabilization ponds as a system to handle waste from septic tanks exhausted. A series of lakes studied belong to one of the largest clean-pit of Natal, consisting of two anaerobic ponds, one facultative and maturation, and a tank disinfection, the wastewater being released in the Potengi River. Samples were collected between the months of October 2007 to October 2008, at six points previously defined and judged as more appropriate to what is proposed study. The analysis results in field and laboratory showed the most significant removal of COD (88.93%), total suspended solids (94.87%), organic nitrogen (66.87%) and thermotolerant coliforms (99.88%). Some results have not reached the expected because the system under study had operating problems that have undermined the efficiency of the reactors

Caracterização do conteúdo de fossas e tanques sépticos na cidade de Natal

Only 32% of the population of Natal is attended by sewage, while the remaining population use pits and septic tanks. The characterization of the contents of septic tanks and pits contributes to the performance of such system and may guide the decision on treatment of these contents. The main of this research is to characterize the contents of interior residential pits and septic tanks in the greater Natal, with the following specific goals: to develop and manufacture a sampler capable of collecting a representative sample of the entire column (the surface scum, the clarified liquid and sludge bottom); to compare the contents of the tanks with the pits; to compare the contents of the septage from vacuum trucks; to relate the composition content with socioeconomic characteristics of households; to compare the content in both chambers of the septic tanks in series; to assess the situation of the content before and six months after the cleanness; and ultimately propose a pilot scale plant for treatment of septage. Once the sampler was developed, samples were collected within 14 septic tanks and 10 pits in many districts of Natal. Medians of the 24 systems were obtained: temperature, pH, conductivity, oil and grease, total solids, total suspended solids and sediments of 28.0 °C, 6.95; 882 mS/cm, 75.2 mg/L; 10,169 mg/L, 6,509 mg/L and 175 mL/L respectively; 111.0 mgN/L for ammonia, 130.5 mgN/ L for organic nitrogen, 0.2 mgN/L for nitrite, 0.4 mg/L for nitrate; 8935 mgO2/L for COD, 29.2 mgP/L for total phosphorus, thermotolerant coliforms from 9.95 E +06 CFU/100mL helminth eggs and 9.2 eggs/L with a maximum concentration of 688 eggs/L and minimum of 0 eggs/L. Medians of organic nitrogen and TKN were significantly different between groups of tanks and pits. The systems with cleanness gap from 11 and 20 years presented the higher concentrations for most variables. The effluent from the toilets and bathrooms participate more effectively in contributing fractions of solids, alkalinity, nitrogen, COD, total phosphorus, thermotolerant coliforms and helminth eggs. The systems used by socioeconomics class with income from R$ 3,700.00 to R$ 7,600.00, presented higher concentrations for COD, nitrogen, solids and helminth eggs. The first of the two chambers had always presented higher concentrations over the second compartment. The analysis of variance for most variables, showed that the values of septic tanks, pits and septage from vacuum trucks belong to the same group. In the samples taken after cleanness, the median of pH and temperature increased, while alkalinity, COD, organic nitrogen, total phosphorus, ammonia and helminth eggs decreased. The oils and greases and thermotolerant coliforms had slightly varied due to the continuous release of sewage into the systems that maintained their steady state concentrations.

Dificuldades no processo ensino aprendizagem de trigonometria por meio de atividades

This study analyses the difficulties that teachers of high school face in the process of the teaching of trigonometry through activities in a construtivist focus. It contains a review of some publications and dissertations related with the study of trigonometry elaborated in the last years by several authors. It resorts to the study of teaching engineering as an instrument used in the research. It also presents a set of activities which will serve as sample to other teachers of mathematics; and points ways for the overcome of the difficulties found

Uma análise dos questionamentos dos alunos nas aulas de números complexos

This work presents a contribution for the studies reffering to the use of the History of Mathematics focusing on the improvement of the Teaching and Learning Process. It considers that the History of Matematics, as a way of giving meaning to the discipline and improve the quality of the Teaching and Learning Process. This research focuses on the questions of the students, classified in three categories of whys: the chronological, the logical and the pedagogical ones. Therefore, it is investigated the teaching of the Complex Numbers, from the questions of the students of the Centro Federal de Educação Tecnológica do Rio Grande do Norte (Educational Institution of Professional and Technology Education from Rio Grande do Norte). The work has the following goals: To classify and to analyse the questions of the students about the Complex Numbers in the classes of second grade of the High School, and to collate with the pointed categories used by Jones; To disccus what are the possible guidings that teachers of Mathematics can give to these questions; To present the resources needed to give support to the teacher in all things involving the History of Mathematics. Finally, to present a bibliographic research, trying to reveal supporting material to the teacher, with contents that articulate the Teaching of Mathematics with the History of Mathematics. It was found that the questionings of the pupils reffers more to the pedagogical whys, and the didatic books little contemplate other aspects of the history and little say about the sprouting and the evolution of methods of calculations used by us as well

A matemática sob a ótica do Tarô: Uma experiência com a educação de jovens e adultos

La recherche de la formation des citoyens critiques et participatifs, dans le travail pédagogique avec les jeunes et les adultes, a besoin d un entraînement pédagogique qui va au delà de l attitude traditionnelle d'apprendre avec des méthodes mécaniques et arbitraires qui, en insistant excessivement sur l image du professeur, donnent priorité à l'enseignement, au détriment de l apprentissage. Dans ce sens, la présente étude, cherchant la possibilité de réalisation d'un travail alternatif pour l'enseignement des Mathématiques, dans une perspective transdisciplinaire, dans le sens de développer l apprentissage significatif des étudiants jeunes et adultes du Projet Croire, présente les résultats d'une recherche-intervention qui a utilisé les lettres du tarot comme ressource didactique en salle de classe. On prétend, avec cela, montrer cet instrument comme facilité d apprentissage de contenus des Mathématiques comme systèmes de numération, nombres entiers et géométrie, en amenant les Mathématiques dans une perspective historique et culturelle et donnant un traitement global à l'acte complexe d'apprendre. Dans ce travail, le jeune étudiant et l étudiant adulte est pris comme individu concret, prenant en considération les aspects cognitifs et les aspects d attitude de son apprentissage, ce qui est favorisé par la nature des lettres du tarot et par la compréhension adoptée, des mathématiques comme système symbolique

Aplicações das atividades estruturadas de Skemp para a construção do conhecimento matemático

The present study constitutes a discussion about the application of Structured Activities to the construction of the mathematical knowledge, proposed by Richard Skemp. The discussion is based on the research that the author carried out in a public school of the state education chain buy using procedures of the research-action. It investigates the possibility of adoption of the proposal of Skemp in a new reality. It utilizes explanations from several theorists to understand the necessity and, at the same time, to enhance the efficiency of the referred activities in first grades of elementary school when students have their first mathematics teachings. It emphasizes the important rule of the teacher, as mediator to the mental constructions of the child. It presents considerations about the results achieved by the research, noticing the possibility of adoption of the studied proposal even though it is necessary an adjustment of the procedures to appropriate didactic-pedagogic requirements to the educational reality in which this project was done

A difícil aceitação dos números negativos: um estudo da teoria dos números de Peter Barlow (1776-1862)

The present study seeks to present a historico-epistemological analysis of the development of the mathematical concept of negative number. In order to do so, we analyzed the different forms and conditions of the construction of mathematical knowledge in different mathematical communities and, thus, identified the characteristics in the establishment of this concept. By understanding the historically constructed barriers, especially, the ones having ontologicas significant, that made the concept of negative number incompatible with that of natural number, thereby hindering the development of the concept of negative, we were able to sketch the reasons for the rejection of negative numbers by the English author Peter Barlow (1776 -1862) in his An Elementary Investigation of the Theory of Numbers, published in 1811. We also show the continuity of his difficulties with the treatment of negative numbers in the middle of the nineteenth century

O uso pedagógico de uma seqüência didática para a aquisição de algumas idéias relacionadas ao conceito de números complexos

The aim of the present work is to contribute to the teaching-learning process in Mathematics through an alternative which tries to motivate the student so that he/she will learn the basic concepts of Complex Numbers and realize that they are not pointless. Therefore, this work s general objective is to construct a didactic sequence which contains structured activities that intends to build up, in each student s thought, the concept of Complex Numbers. The didactic sequence is initially based on a review of the main historical aspects which begot the construction of those numbers. Based on these aspects, and the theories of Richard Skemp, was elaborated a sequence of structured activities linked with Maths history, having the solution of quadratic equations as a main starting point. This should make learning more accessible, because this concept permeates the students previous work and, thus, they should be more familiar with it. The methodological intervention began with the application of that sequence of activities with grade students in public schools who did not yet know the concept of Complex Numbers. It was performed in three phases: a draft study, a draft study II and the final study. Each phase was applied in a different institution, where the classes were randomly divided into groups and each group would discuss and write down the concepts they had developed about Complex Numbers. We also use of another instrument of analysis which consisted of a recorded interview of a semi-structured type, trying to find out the ways the students thought in order to construct their own concepts, i.e. the solutions of the previous activity. Their ideas about Complex Numbers were categorized according to their similarities and then analyzed. The results of the analysis show that the concepts constructed by the students were pertinent and that they complemented each other this supports the conclusion that the use of structured activities is an efficient alternative for the teaching of mathematics

Estudo histórico e pedagógico sobre ternos pitagóricos à luz de eugène bahier

At the present investigation had the purpose to achieve a descritive analysis pedagogy in the work of Recherche méthodique et propriétés des triangles rectangles en nombres entiers. According to the analysis achieved, we made and applyed the teaching module called Pitagories: one of tools to comprehension Pitagory Theorema, there were studying by public students in mathematic course in the UFRN , the new mathematic teachers in future. The analysis the was made with writen test the was showed that all students got the view comprehension in the teaching approach module, to apointed the difference in the learning qualytative with other reseach that was made with quastionaire and enterview. With this module that was made with the new future teacheres there was more attention the better comprehension with the Pitagory Theorema, that was good focus in the pitagory about the potential historical pedagogyc in the work studied.

Atribuição de significado ao conceito de proporcionalidade: contribuições da história da matemática

This study is the result of a work which approaches the Mathematics History how source of the meaning s attribution in the proportionality concept. We adopt the methodology of the source qualitative and we work with a group of teachers from instruction s public system of the fundamental and medium level from Pocinhos City Paraíba. For the data collection, we use the field notes, the questionnaire, a sequence of activities and the interview semistructured like instruments. The study had how objective to know the significates attributeds to proportionality concept through of the activity mediate from Mathematics History, besides to investigate if a approach of the nature enables modification according to this sense. The results obtaineds though the data analysis indicate that the activities bring contributions which refer to achieve objectives. On the other hand they also showed that we have a long trajectory to be trailed in the meaning of to turn the Mathematics History a subsidy effective in the teachers practice, in view of the formation absence in the knowledge area, besides the necessity of the approach adequated of the Mathematics History in the didatics books of Mathematic

Um estudo sobre o ensino-aprendizagem das demonstrações matemáticas

Demonstrations are fundamental instruments for Mathematics and, as such, are frequently used by mathematicians, math teachers and students. In fact, demonstrations are part of every Mathematics teaching environment, because Mathematics considers something true when it can be demonstrated. This is in contrast to other fields of knowledge that employ observation and experimentation to validate truth. This dissertation presents a study of the teaching and learning of demonstrations in Mathematics, describing a Teaching Module applied in a course on the Theory of Numbers offered by the Mathematics Department of the Universidade Federal do Rio Grande do Norte for mathematics majors. The objective of the dissertation was to propose and test a Teaching Module that can serve as a model for teaching demonstrations. The Teaching Module consisted of the following five steps: the application of a survey to determine the students‟ profiles and their previous knowledge of mathematical language and techniques of demonstration; the analysis of a series of dialogues containing arguments in everyday language; the investigation and analysis of the structure of some important techniques of demonstration; a written assessment; and, finally, an interview to further verify the principal results of the Teaching Module. The analysis of the data obtained though the classroom activities, written assessments and interviews led to the conclusion that there was a significant amount of assimilation of the issue at the level of relational understanding, (SKEMP, 1980). These instruments verified that the students attained considerable improvement in their use of mathematical language and of the techniques of demonstration presented. Thus, the evidence supports the conclusion that the proposed Teaching Module is an effective means for the teaching/learning of mathematical demonstration and, as such, provides a methodological guide which may lay the foundations for a new approach to this important subject