Ratio and Proportion

Esta ponencia surge en el marco de una investigacion que busca explorar los efectos del diseno de una secuencia de tareas, para apoyar el desarrollo del pensamiento proporcional, de un grupo de estudiantes de grado septimo de la educacion basica en Bogota. Dentro de dicho estudio, se han revisado las perspectivas teoricas de las razones, proporciones y proporcionalidad (RPP), para fortalecer el saber matematico de los investigadores y su conocimiento pedagogico del contenido. Emergen asi: la Antifairesis o Antanairesis, la proporcionalidad abordada desde la obra Elementos de Euclides (Libros V y VII), la teoria de las magnitudes para la proporcionalidad y la teoria de la proporcionalidad centrada en los numeros reales. El enfoque para presentar estas miradas de las RPP es el de relacion de comparacion, porque la aritmetizacion como cociente de la razon y la proporcion en la ensenanza, desvirtua su aspecto mas fundamental, vincular dos objetos matematicos.

Esta ponencia surge en el marco de una investigacion que busca explorar los efectos del diseno de una secuencia de tareas, para apoyar el desarrollo del pensamiento proporcional, de un grupo de estudiantes de grado septimo de la educacion basica en Bogota. Dentro de dicho estudio, se han revisado las perspectivas teoricas de las razones, proporciones y proporcionalidad (RPP), para fortalecer el saber matematico de los investigadores y su conocimiento pedagogico del contenido. Emergen asi: la Antifairesis o Antanairesis, la proporcionalidad abordada desde la obra Elementos de Euclides (Libros V y VII), la teoria de las magnitudes para la proporcionalidad y la teoria de la proporcionalidad centrada en los numeros reales. El enfoque para presentar estas miradas de las RPP es el de relacion de comparacion, porque la aritmetizacion como cociente de la razon y la proporcion en la ensenanza, desvirtua su aspecto mas fundamental, vincular dos objetos matematicos.

The aim of this intervention study is to improve secondary school students’ ability of thinking in terms of functions with regard to the aspect of covariation. The aspect of covariation focuses on a particular aspect of the concept of function: the direction and scale of the change of the y-value induced by a certain increase of the x-value. In Germany, the aspect of covariation is usually taught in grades 11-13 when dealing with infinitesimal calculus where the derivative is introduced as the instantaneous rate of change. From a more general point of view covariational thinking can be considered as a sub-category of dynamical thinking. Dealing with problems where dynamical thinking is involved, the person will have to predict the status of a system if a certain change in the system occurs (Ossimitz 2000). The intention of this poster (70 cm x 90 cm) is to display the training sessions and the design of the study.

One can search for information online in many sources and never arrive at a satisfactory explanation of what a percentage means. In this short article I teach you exactly what is a percentage and what children need to know before they can fully understand the concept.

To conclude, the word ‘ratio’ literally means the comparison of two magnitudes, and whether these are in proportion or not: that is, having the property of quotientness (or common divisor).

Συμπερασματικά, η λέξη «λόγος» σημαίνει κυριολεκτικά τη σύγκριση δύο μεγεθών, και κάτα ποια σχέσις είναι «αναλογία»: δηλαδή ιδιότητα πηλικότητάς.

Este trabalho enfoca memórias, conotações, discursos e crenças que gravitam em torno dos intervalos de três tons: a quarta aumentada e a quinta diminuta. A problemática parte da versada noção de que o trítono foi proibido durante a Idade Média pela Igreja por sua associação com o diabo, noção epitomada pelo ditado mi contra fa est diabolus in musica [mi contra fa é o diabo na música].

Background The COVID-19 pandemic had a substantial impact on mathematics teaching and learning in South African schools in 2020 and 2021, particularly in poor communities where online learning was not possible. This loss of learning opportunities is likely to have knock-on effects for several years to come because of the hierarchical nature of mathematics. In the last quarter of 2020, the Wits Maths Connect Secondary (WMCS) project, in partnership with the Gauteng Department of Education (GDE) and Olico Maths Education, developed and piloted a baseline assessment of Grade 7 learners' mathematical knowledge to provide an indicator of what mathematics Grade 7 learners were bringing to high school at the start of the 2021 school year. Given the dual purpose of the test as both a diagnostic and a baseline measure, the authors coined the name DiBa Test. In addition, they were confident that an appropriately designed test instrument would provide diagnostic evidence of learners' errors which could, in turn, inform teaching. In the longer term, they consider the development of such a test instrument to be strategic for wider use provincially and nationally. A group of Grade 8 learners were included in the pilot to compare with the Grade 7s. This stemmed from concerns that Grade 8s had very limited learning opportunities as a result of school closures from mid-March to late August 2020. By contrast, Grade 7s had returned to school in June 2020, although schools were then closed for another month in late July. Based on the reduced opportunity to learn for Grade 8s, the authors wanted to investigate possible differences in performance across the two groups, suspecting that Grade 7s might outperform the Grade 8s on some items. Background: Poor mathematics performance in South Africa is well known. The COVID-19 pandemic was expected to exacerbate the situation. Aim: To investigate Grade 7 learners' mathematical knowledge at the end of primary school and to compare mathematical performance of Grade 7 and 8 learners in the context of the pandemic. Setting: Data were collected in term four of 2020 at 11 primary schools and five secondary schools. All schools drew learners from poor communities in Gauteng. Methods: A multiple-choice test covering mathematical content from Grades 4-7 was designed and piloted. Learner performance was measured through number of correct responses. Qualitative error analyses were conducted on learners' choices of distractors. Results: The difference in performance of the two grade groups was not statistically significant. There were similar response patterns in learners' choices of distractors with strong evidence of cue-based reasoning and evidence of additive reasoning in items requiring multiplicative reasoning. Conclusion: Grade 8 learners made very small gains, likely due to reduced learning time. Learner errors show many similarities with the international literature and show that Grade 7 learners are not yet ready for algebra. Contribution: The findings provide starting points for addressing the most common errors and highlight the need for: greater attention to whole and rational number concepts in Grade 8; strategies for teacher support in teaching primary maths content; and innovative teaching strategies to fast-track learning of this content.

El presente trabajo de investigación aborda los razonamientos que tienen los estudiantes del grado quinto sobre el concepto de proporcionalidad, según el modelo educativo de van Hiele; por lo tanto, el estudio permitió establecer unos descriptores de nivel que caracterizaron y determinaron el nivel de razonamiento en que se encontraban razonando dichos estudiantes. Este estudio se desarrolló en el marco del programa de Maestría en Educación, en la línea de educaciónmatemática, que se desarrolla en una de las sedes regionales de la Universidad de Antioquia, con la participación del grupo de investigación EDUMATH.

Esta comunicacion breve da a conocer y pone en consideracion de la comunidad de Educadores Matematicos, un posible camino para abordar las operaciones con fracciones, haciendo uso de unidad de medida comun fraccionaria. La propuesta es continuacion al trabajo de grado realizado en el marco de la especializacion en Educacion Matematica de la Universidad Distrital Francisco Jose de Caldas. La experiencia se desarrollo con estudiantes de grado octavo en el 2006. Tiene como pretension sumergirse en el mundo de las fracciones para operar con ellas a traves de tomarlas como unidad de medida y el establecimiento de unidad de medida comun fraccionaria entre dos o mas fracciones unitarias.

A continuación se detalla una experiencia de aula realizada en el primer semestre del año 2014 en el colegio Nueva Esperanza, que da cuenta de la construcción de esquemas Splitting en grado quinto, generando una propuesta transistémica pretendida por los Lineamientos Curriculares (1998) por el Ministerio de Educación Nacional

The Malay cultural landscape has become an integral part of examining the cultural significance in Malaysia. Thus, the main objective of the study is to determine the impact of the visual characteristics of the built cultural landscape in conserving and preserving Kota Bharu as an aristocratic area in this modern era. Kota Bharu is one of the Islamic cities in Malaysia. This study embarks into a quantitative study using a mixed-method. A significant implication is to propose a significant model to value the Malay cultural heritage and preserve the culture and spirit among the Kelantanese Malay population who live in the modern era.

Resumen. El presente documento tiene por objetivo evidenciar algunos de los significados asociados a la noción de fracción al trabajar con un problema de mezclas dentro del aula. La escuela elegida para su realización pertenece al sistema educativo estatal del nivel básico,

Núcleo temático: I Enseñanza y aprendizaje de la matemática en las diferentes modalidades y niveles educativos. Modalidad: CB Nivel educativo: Bachillerato Palabras clave: Ilusión de la linealidad, bachillerato, resolución de problemas, diagnóstico. Resumen El presente avance de investigación muestra una visión general de las tendencias de los alumnos de bachillerato al resolver problemas de tipo constante, área, volumen y con falta de autenticidad donde se hace presente la ilusión de la linealidad. Uno de los ejemplos más comunes de un comportamiento corrompido en la resolución de problemas matemáticos es la fuerte tendencia de los alumnos a aplicar métodos proporcionales a los problemas de valor faltante, incluso en problemas en los que es cuestionable o claramente inadecuado (De Bock, Van Dooren, Janssens & Verschaffel, 2007). Se aplicó un instrumento con el cual se pudo observar que la mayoría de los alumnos son "atrapados" por la ilusión de la linealidad pues tienden a generalizar en problemas de área y volumen debido a que suponen que si en determinada figura su arista crece k-veces entonces su área o volumen también crece k-veces. Otros resultados que se encontraron fueron que los alumnos ignoran consideraciones realistas o no toman en cuenta algunos aspectos esenciales de la situación del problema en la vida real. Bastantes alumnos realizaron representaciones externas, sin embargo fueron de poca ayuda para la interpretación y resolución correcta de los problemas. Introducción Las reformas y programas de estudio de varios países tienen como uno de los principales objetivos de la educación matemática propiciar la capacidad de desarrollar y utilizar modelos para dar sentido a las diversas situaciones que rodean la vida diaria y de los sistemas complejos derivados de nuestra sociedad moderna (Blum,

The earlier masjid architectural styles were similar to Malay vernacular houses and evolved with the introduction of the modernist style. This resulted in increased energy consumption. Hence, a proper set of thermal comfort is important to provide a suitable environment for the masjid’s occupants. The vernacular characteristics in the passive designs have caused designers to find solutions towards sustainable designs. Therefore, designing a good thermal performance building can be done by readapting the Malay vernacular architecture and passive design strategies for modern masjids. This paper is aimed to document the literature and potential case studies in identifying the relationship between the design of Malay vernacular and thermal comfort elements in a hot and humid climate. The outcome is to formularize requirements of thermal comfort in Malay vernacular masjids based on four major factors namely i) architectural qualities, ii) construction systems, iii) structural components ...

Resumen. El presente documento tiene por objetivo evidenciar algunos de los significados asociados a la noción de fracción al trabajar con un problema de mezclas dentro del aula. La escuela elegida para su realización pertenece al sistema educativo estatal del nivel básico,

This study aims to describe students’ proportional reasoning profile of junior high school students in mathematical problem solving based on Adversity Quotient (AQ) type climbers, campers, and quitters. This qualitative research was conducted to 22 female students who were in eight grade that used documentation, test, and interview to gather the data. Analyzing the students’ test result and then interviewing them for each category were done for the analysis process. The result show that there are 9.1% climber students, 72.7% camper students, and 18.2% quitter students. The result of this study showed that there were differencies about proportional reasoning activities based on type Adversity Quotient (AQ) climbers, campers, and quitters.

Núcleo temático: I Enseñanza y aprendizaje de la matemática en las diferentes modalidades y niveles educativos. Modalidad: CB Nivel educativo: Bachillerato Palabras clave: Ilusión de la linealidad, bachillerato, resolución de problemas, diagnóstico. Resumen El presente avance de investigación muestra una visión general de las tendencias de los alumnos de bachillerato al resolver problemas de tipo constante, área, volumen y con falta de autenticidad donde se hace presente la ilusión de la linealidad. Uno de los ejemplos más comunes de un comportamiento corrompido en la resolución de problemas matemáticos es la fuerte tendencia de los alumnos a aplicar métodos proporcionales a los problemas de valor faltante, incluso en problemas en los que es cuestionable o claramente inadecuado (De Bock, Van Dooren, Janssens & Verschaffel, 2007). Se aplicó un instrumento con el cual se pudo observar que la mayoría de los alumnos son "atrapados" por la ilusión de la linealidad pues tienden a generalizar en problemas de área y volumen debido a que suponen que si en determinada figura su arista crece k-veces entonces su área o volumen también crece k-veces. Otros resultados que se encontraron fueron que los alumnos ignoran consideraciones realistas o no toman en cuenta algunos aspectos esenciales de la situación del problema en la vida real. Bastantes alumnos realizaron representaciones externas, sin embargo fueron de poca ayuda para la interpretación y resolución correcta de los problemas. Introducción Las reformas y programas de estudio de varios países tienen como uno de los principales objetivos de la educación matemática propiciar la capacidad de desarrollar y utilizar modelos para dar sentido a las diversas situaciones que rodean la vida diaria y de los sistemas complejos derivados de nuestra sociedad moderna (Blum,

The woodcarving have considered as an important component in traditional Malay vernacular architecture were meant to perform visual values. This study examined of the wall woodcarving decoration, and it motifs found in traditional Malay house with two research questions. First, does the decoration and motifs influences social interaction of spatial? Second, is it the choice of decoration and motifs applied on the wall of Malay traditional house are intentional or accidental? This study can encourage in the direction of quality of life and sustainability of Malay's heritage, also identifies best-practices of choosing the proper woodcarving design to the interior space category in the future.

In this article we explore approaches to curriculum in the primary school in order to map and manage the omissions implicit in the current unfolding of the Curriculum and Assessment Policy Statement for mathematics. The focus of school-based research has been on curriculum coverage and cognitive depth. To address the challenges of teaching mathematics from the perspective of the learner, we ask whether the learners engage with the subject in such a way that they build foundations for more advanced mathematics. We firstly discuss three approaches that inform the teaching of mathematics in the primary school and which may be taken singly or in conjunction into organising the curriculum: the topics approach, the process approach, and the conceptual fields approach. Each of the approaches is described and evaluated by presenting both their advantages and disadvantages. We then expand on the conceptual fields approach by means of an illustrative example. The planning of an instructional ...

In the transition years, Grades 7 to 9, the shift from natural numbers to rational numbers and the associated multiplicative concepts prove challenging for many learners. The new concepts, operations and notation must be mastered if the student is to thereafter rise to meet the challenges of algebra and more advanced and powerful mathematics. The multiplicative conceptual field (MCF) groups together such concepts as fraction, ratio, rate, percentage and proportion, all of which are related yet subtly distinct from one another, each with its own challenges. Rasch analysis allows us to compare the difficulty of mathematical problems located within the MCF while, on the same scale, locating the degree to which individual learners have mastered the necessary skill set. Such location of problems and learners on the same unidimensional scale allows for fine-grained analysis of which aspects of the problems being analysed make one problrm more difficult than another. Simultaneously the scale gives the teacher clear evidence of which students have mastered which concepts and skills and which have not, thereby allowing more targeted assistance to the class and individual learners. This paper illustrates the process involved in such analysis by reporting on results located within a larger study. It is suggested that implementing Rasch analysis within the school classroom on appropriately designed assessment instruments would provide clarity for the teacher on the locations of difficulty within the problems used in the assessment and the relative degree to which individual learners are achieving success at mastering the targeted concepts.

The rationale for the introduction of standards in the United States in the late 1980s was that the quality of education would improve. Assessment instruments in the form of written tests were constructed in order to perform a monitoring function. The introduction of standards and the associated monitoring have been replicated in South Africa. It was intended that these elements would result in a more equitable education across the country. In neither of these countries has this process borne the results expected. The lack of substantive progress may be due to the fact that socioeconomic disadvantage and, therefore, opportunity to learn, has not been addressed. It may also be that systemic-type assessments have little meaning for the teachers, and bear little relation to classroom assessments; the perceived lack of relevance resulting in a lack of commitment to the process. Our education system is in need of serious deliberations about the broad curriculum goals relevant to society, a classroom environment that is conducive to learning and an assessment model that supports instruction. Within the assessment model we propose an instrument design that supports instruction. The assessment model includes a monitoring component, a formative component and a professional development component. We propose an assessment process where general trends can be reported for systemic purposes but also that the results of the individual learner progress obtained through both a monitoring and a formative component, are to be suitable for engagement by teachers. We honour the central teacher role in communicating both the emerging teaching successes and the currently troublesome areas of classroom learning challenges.

The purpose of this study is to apply the Rasch model to investigate both the Van Hiele theory for geometric development and an associated test. In terms of the test, the objective is to investigate the functioning of a classic 25-item instrument designed to identify levels of geometric proficiency. The data set consists of responses by 244 students (106 for a pre-test and 138 for a post-test) of which 76 students sat both the pre-test and the post-test. The summary item statistics do not show statistically discernible differences between observed and expected scores under the Rasch model (Chi-square statistic). The Rasch analysis confirms to some strong extent the Van Hiele theory of geometric development. The study identifies some problematic test items as they only require knowledge of a specific aspect of geometry instead of testing geometric reasoning. In terms of the Van Hiele theory, the Rasch analyses identified as problematic some items about class inclusion, an issue which has also been raised in other studies.

The developments in mathematics that take place in Grades 7 to 9 constitute critical nodes in a learner’s scholing. One of the major transitions to be made is from whole numbers to real numbers, which involves theunderstanding of rational (and irrational) numbers, and concepts such as ratio, proportion and percent. I hypothesise that ratio is a threshold concept that provides the conceptual gateway to higher order concepts. The research problem is to describe the learning challenges and provide an array of insightsand strategies that will inform teaching. The theory of conceptual fields provides the framework for this research (Vergnaud, 1988). The Rasch measurement model (Rasch, 1960/1980) articulates the qualitative and quantitative aspects of the research. This paper provides an overview of the broader study and reports on an aspect of the data analysis.

Núcleo temático: I Enseñanza y aprendizaje de la matemática en las diferentes modalidades y niveles educativos. Modalidad: CB Nivel educativo: Bachillerato Palabras clave: Ilusión de la linealidad, bachillerato, resolución de problemas, diagnóstico. Resumen El presente avance de investigación muestra una visión general de las tendencias de los alumnos de bachillerato al resolver problemas de tipo constante, área, volumen y con falta de autenticidad donde se hace presente la ilusión de la linealidad. Uno de los ejemplos más comunes de un comportamiento corrompido en la resolución de problemas matemáticos es la fuerte tendencia de los alumnos a aplicar métodos proporcionales a los problemas de valor faltante, incluso en problemas en los que es cuestionable o claramente inadecuado (De Bock, Van Dooren, Janssens & Verschaffel, 2007). Se aplicó un instrumento con el cual se pudo observar que la mayoría de los alumnos son "atrapados" por la ilusión de la linealidad pues tienden a generalizar en problemas de área y volumen debido a que suponen que si en determinada figura su arista crece k-veces entonces su área o volumen también crece k-veces. Otros resultados que se encontraron fueron que los alumnos ignoran consideraciones realistas o no toman en cuenta algunos aspectos esenciales de la situación del problema en la vida real. Bastantes alumnos realizaron representaciones externas, sin embargo fueron de poca ayuda para la interpretación y resolución correcta de los problemas. Introducción Las reformas y programas de estudio de varios países tienen como uno de los principales objetivos de la educación matemática propiciar la capacidad de desarrollar y utilizar modelos para dar sentido a las diversas situaciones que rodean la vida diaria y de los sistemas complejos derivados de nuestra sociedad moderna (Blum,

One of the students' reasoning ability need to be developed is proportional reasoning abilities as the basic reasoning of the five types of reasoning that have been expressed by Piaget. This types of research is a qualitative descriptive research that aims to categorize first middle class students' proportional reasoning abilities in solving pre-multiplicative problems. This reseach used purposive sampling technique, namely the selection of subjects with certain considerations to get subjects who have different categories of proportional reasoning. The instrument was a written test of a three point description and guiding interview. Then, the analysis activity used Miles and Huberman models of data reduction, data display, and conclusion drawing/verification. Based on the results of the research, it can be concluded that the development of proportional reasoning abilities mastered by students is different, namely subject 1 reaches category 3, subject 2 reaches category 1 and subject 3 reaches category 0. Abstrak: Salah satu kemampuan bernalar siswa yang perlu dikembangkan adalah kemampuan penalaran proporsional sebagai penalaran dasar dari kelima jenis penalaran yang telah diungkapkan oleh Piaget. Jenis penelitian ini merupakan penelitian deskriptif kualitatif yang bertujuan untuk mengkategorikan kemampuan penalaran proporsional siswa kelas menengah pertama dalam menyelesaikan masalah pra-multiplikatif. Penelitian ini menggunakan teknik Purposive Sampling yaitu pemilihan subjek dengan pertimbangan tertentu untuk mendapatkan subjek yang memiliki kategori penalaran proporsional berbeda. Instrumen yang digunakan adalah tes tertulis berupa uraian berjumlah tiga soal dan pedoman wawancara. Kemudian dilanjutkan dengan kegiatan analisis data menggunakan model Miles dan Huberman yang terdiri dari reduksi data, penyajian data, dan penarikan kesimpulan. Berdasarkan hasil penelitian, dapat disimpulkan bahwa perkembangan kemampuan penalaran proporsional yang dikuasai siswa berbeda-beda yaitu subjek 1 mencapai kategori 3, subjek 2 mencapai kategori 1 dan subjek 3 mencapai kategori 0.

The serambi is a pronounced interior space within the Malays traditional home design, particularly for the Negeri Sembilan Malays in Rembau, the context of this exploratory case study research. This paper presents the interpretations of cultural values and socio-cultural viewpoint on the quality of life derived from critical, comprehensive literature reviews within the serambi context. 'Cultural values' concluded to be the infused refinement of the Adat Perpatih, through the inculcation of the custom in all aspect of serambi design. The local Malay community's socio-cultural viewpoint on the quality of life, therefore, theorized as the inclusion of cultural values within their home design.

The woodcarving have considered as an important component in traditional Malay vernacular architecture were meant to perform visual values. This study examined of the wall woodcarving decoration, and it motifs found in traditional Malay house with two research questions. First, does the decoration and motifs influences social interaction of spatial? Second, is it the choice of decoration and motifs applied on the wall of Malay traditional house are intentional or accidental? This study can encourage in the direction of quality of life and sustainability of Malay's heritage, also identifies best-practices of choosing the proper woodcarving design to the interior space category in the future.

This paper presents a review on holistic approaches of green building assessment tools (GBAT) for sustainable development (SD) showing the trends and conceptual framework. The method of the study is through literature review which highlighted the socio-cultural inadequacy of most GBAT. The paper proposes several hypotheses. Firstly, to use a holistic universal method to assess sustainability within the community's cultural context. Secondly, the assessment criteria for sustainability from the socio-cultural viewpoint would differ from the conventional tools. Thirdly, the study proposes that the community would prefer to shape their future environment with specific preferred values in their home environment.

This communication presents a study of university students' extension of linear models to non-linear situations. By linear models we mean the model ya.x+b, particular representations of direct proportionality and the diagram for the rule of three. This phenomenon has been studied at the primary and secondary levels, and has been called "illusion of linearity" or "linear misconception" (Behr, Hare, Post, & Lesh, 1992; De Bock, Von Doorem, Janssens, & Verschaffel, 2002; De Bock, Von Doorem, Verschaffel, & Janssens, 2001; De Bock, Verschaffel & Janssens, 1998). There exists agreement in describing the "linear misconception" as persistent and resistant to change. Studies about this phenomenon among university students are not frequent, even though its presence and persistence have been observed in diverse types of problems and contexts at that level. That situation led us to carry out an exploratory study to document, describe and analyze the presence ...

The challenges inherent in assessing mathematical proficiency depend on a number of factors, amongst which are an explicit view of what constitutes mathematical proficiency, an understanding of how children learn and the purpose and function of teaching. All of these factors impact on the choice of approach to assessment. In this article we distinguish between two broad types of assessment, classroom-based and systemic assessment. We argue that the process of assessment informed by Rasch measurement theory (RMT) can potentially support the demands of both classroom-based and systemic assessment, particularly if a developmental approach to learning is adopted, and an underlying model of developing mathematical proficiency is explicit in the assessment instruments and their supporting material. An example of a mathematics instrument and its analysis which illustrates this approach, is presented. We note that the role of assessment in the 21st century is potentially powerful. This infl...

Una de las principales causas de las dificultades de los estudiantes de educación primaria y secundaria con las operaciones con los números racionales se debe al uso inapropiado de su conocimiento de los números naturales. Este fenómeno es conocido como natural number bias. Nuestra investigación tiene como objetivo examinar los niveles de éxito y los razonamientos de los estudiantes desde 5.º de educación primaria hasta 4.º de educación secundaria cuando resuelven tareas de multiplicación de un número natural por un racional. Los participantes fueron 438 estudiantes españoles de educación primaria y secundaria. Los resultados muestran porcentajes de éxito menores en tareas donde el conocimiento de los números naturales no es compatible para resolverlas. El análisis de los razonamientos de los estudiantes en estas tareas evidencia la existencia del fenómeno natural number bias en educación primaria y secundaria, pero mostrando su disminución en los últimos años. Estos resultados ampl...

This study focused on 26 girls' development of proportional reasoning in two fifthgrade classrooms in Iceland. The students were used to instructional practices that encouraged them to devise their own solutions to mathematical problems. The results supported four levels of proportional reasoning. Level 1, girls showed limited ratio knowledge. Level 2, they perceived the given ratio as an indivisible unit. Level 3, students conceived of the given ratio as a reducible unit. And at Level 4 students no longer thought of ratios exclusively as unit quantities, but understood the proportion in terms of multiplicative relations. The results suggest that students can reach level 3 reasoning with less struggle than it takes to achieve level 4, which suggests that the knowledge needed to operate on level 3 was within their reach. OBJECTIVES This study investigates the developmental of proportional reasoning of girls in two fifth-grade classes in Iceland. The purposes of this study was to further investigate four levels of proportional reasoning identified in a pilot study that the author conducted in collaboration prior to the study reported here (Carpenter et al. 1999). In particular, do the four levels describe the pathway of a population of Icelandic girls before, during, and after they have engaged in a unit focused on proportional reasoning? Secondly, what evidence is there for the existence of Level 2, Level 3, and Level 4 ways of reasoning in students' verbal protocols? And finally how does instruction that is focused on students' reasoning help students make the transition from level to level? BACKGROUND AND THEORETICAL ORIENTATION Proportional reasoning represents a cornerstone in the development of children's mathematical thinking (Inhelder & Piaget, 1958; Resnick & Singer, 1993). Ratio and proportion are critical ideas for students to understand; however, although young children demonstrate foundations for proportional reasoning, students are slow to attain mastery of these concepts. Many studies on children's proportional reasoning provide evidence of various influences on students' thinking about proportion. Among these influential factors are the problem numerical structures 1. The number structure refers to the multiplicative relationship within and between ratios in a proportional setting. A "within"

ABSTRAK Hariyanti, Uun. 2016. Penalaran Proporsional dalam Menyelesaikan Masalah Multiplikatif bagi Siswa Kelas VIII SMP. Tesis. Jurusan Pendidikan Matematika, Program Pascasarjana Universitas Negeri Malang. Pembimbing: (I) Dr. Edy Bambang Irawan, M.Pd, (II) Dr. Erry Hidayanto, M.Si. Kata kunci: penalaran, penalaran proporsional, multiplikatif. Penalaran proporsional merupakan penalaran yang penting dalam pembelajaran matematika karena dapat mempengaruhi kemampuan matematis siswa dalam belajar berbagai macam topik yang mengharuskan siswa untuk berpikir kuantitatif dan pemahaman. Tujuan dari penelitian ini adalah untuk mendeskripsikan penalaran proporsional siswa kelas VIII SMPN 21 Malang dalam menyelesaikan masalah multiplikatif. Penelitian ini menggunakan rancangan deskriptif-kualitatif. Peneliti merupakan instrumen kunci (researcher as key instrument). Data yang diperoleh dari penelitian ini adalah (1) hasil validasi, (2) hasil tes siswa dalam mengerjakan tes masalah multiplikatif...

En este trabajo se ensaya la descripción de las praxeologías que se ponen en juego para la enseñanza de los Números Racionales en la escuela primaria (5to- segundo ciclo) y en la escuela secundaria (2do y 3er - ciclo básico) pertenecientes a la Provincia de Buenos Aires. El mismo es parte de un estudio más amplio que fue presentado para acceder al título de Licenciatura en Ciencias de la Educación en Matemática, en la Facultad de Ciencias Exactas de la Universidad Nacional del Centro de la Provincia de Buenos Aires. El conocimiento de las praxeologías permitiría comprender cómo y cuándo se enseñan los Números Racionales en esos niveles educativos y formular hipótesis de las causas de las dificultades encontradas en el aprendizaje de estos saberes por parte de los alumnos. Además, la descripción de las praxeologías se convertiría en un punto de partida para generar otras investigaciones que profundicen este, u otros, aspectos vinculados a la enseñanza y aprendizaje de los Números Rac...

One of the main purposes of the research project that is being carried out in Mexico consists on building, from Filloy's point of view, a Local Theoretical Model (LTM) for fractions and rational numbers. For this purpose, four components of the LTM have to be constructed: models of formal competence, teaching models, models for the cognitive processes and models of communication. In this paper, a network of notions, concepts and processes for fractions and rational numbers that constitutes a fundamental part of the models of formal competence is described. This network was structured taking into account five mathematical processes grounded on Freudenthal's specimen of Didactical Phenomenology of fractions.

El artículo expone el análisis de las representaciones de doce participantes respecto del cálculo operacional al manipular las variables de velocidad, tiempo y distancia y su relación, en la solución de un problema de proporcionalidad presentado por el Software URANUS v1 y tres tareas de control. Se utilizó el método microgenético para analizar los patrones de ejecución y entrevistas semiestructuradas para inferir por análisis de contenido de las verbalizaciones, las representaciones respectivas. Se ilustran tres casos, novato, intermedio y experto, quienes utilizaron diversos cálculos de adición, multiplicación y proporcionalidad y distintas relaciones entre variables, desde la ausencia de nexos, hasta concebirlas interrelacionadas respecto de la distancia. Cinco sujetos intermedios presentaron la representación tipo R3 de cálculos aditivos con relación entre las variables que produce efectos. Un experto reveló la representación tipo R5 de cálculo proporcional. En síntesis, la solu...

El artículo expone el análisis de las representaciones de doce participantes respecto del cálculo operacional al manipular las variables de velocidad, tiempo y distancia y su relación, en la solución de un problema de proporcionalidad presentado por el Software URANUS v1 y tres tareas de control. Se utilizó el método microgenético para analizar los patrones de ejecución y entrevistas semiestructuradas para inferir por análisis de contenido de las verbalizaciones, las representaciones respectivas. Se ilustran tres casos, novato, intermedio y experto, quienes utilizaron diversos cálculos de adición, multiplicación y proporcionalidad y distintas relaciones entre variables, desde la ausencia de nexos, hasta concebirlas interrelacionadas respecto de la distancia. Cinco sujetos intermedios presentaron la representación tipo R3 de cálculos aditivos con relación entre las variables que produce efectos. Un experto reveló la representación tipo R5 de cálculo proporcional. En síntesis, la solu...

This communication presents a study of university students' extension of linear models to non-linear situations. By linear models we mean the model ya.x+b, particular representations of direct proportionality and the diagram for the rule of three. This phenomenon has been studied at the primary and secondary levels, and has been called "illusion of linearity" or "linear misconception" (Behr, Hare, Post, & Lesh, 1992; De Bock, Von Doorem, Janssens, & Verschaffel, 2002; De Bock, Von Doorem, Verschaffel, & Janssens, 2001; De Bock, Verschaffel & Janssens, 1998). There exists agreement in describing the "linear misconception" as persistent and resistant to change. Studies about this phenomenon among university students are not frequent, even though its presence and persistence have been observed in diverse types of problems and contexts at that level. That situation led us to carry out an exploratory study to document, describe and analyze the presence ...

The concept of fraction has motivated the carrying out of various studies in the field of Mathematics Education, being a priority research topic, the results of which have contributed to various curricular modifications. This article presents a bibliographic review on the teaching and learning of the fraction concept in different contexts within the research in Mathematics Education. The study was descriptive and consisted of 3 phases: search, organization and analysis of the information. It was evidenced that the understanding of the concept of fraction must be strengthened from an early age since its appropriation can have implications in the personal and professional performance of an individual. It is concluded that the subject of fractions should be widely studied, because the literature shows gaps in its teaching and in the mechanisms that ensure learning, as well as weaknesses in the understanding that both teachers and students express of the concept.El concepto de fracción ...

El taller emana de las discusiones, reflexiones y avances conceptuales generados al interior del grupo de investigación que indaga por los “Modelos y Prácticas Evaluativas de las Matemáticas en la Educación Básica: El caso del Campo Multiplicativo”. Los resultados de investigación cognitiva de algu- na manera cuestionan la forma como se presentan en el currículo los conceptos matemáticos (como conceptos atomizados), además que aportan para rupturar la idea que la adquisición de conceptos pue- de continuar siendo considerada lineal y que es asunto de presentarlos como listado de temas

El artículo expone el análisis de las representaciones de doce participantes respecto del cálculo operacional al manipular las variables de velocidad, tiempo y distancia y su relación, en la solución de un problema de proporcionalidad presentado por el Software URANUS v1 y tres tareas de control. Se utilizó el método microgenético para analizar los patrones de ejecución y entrevistas semiestructuradas para inferir por análisis de contenido de las verbalizaciones, las representaciones respectivas. Se ilustran tres casos, novato, intermedio y experto, quienes utilizaron diversos cálculos de adición, multiplicación y proporcionalidad y distintas relaciones entre variables, desde la ausencia de nexos, hasta concebirlas interrelacionadas respecto de la distancia. Cinco sujetos intermedios presentaron la representación tipo R3 de cálculos aditivos con relación entre las variables que produce efectos. Un experto reveló la representación tipo R5 de cálculo proporcional. En síntesis, la solu...