Ratio and proportion: Research and teaching in mathematics teachers' education (Pre- and in-service mathematics teachers of elementary and middle school classes)
Abstract
Ratio and Proportion-Research and Teaching in Mathematics Teachers' Education offers its readers an intellectual adventure where they can acquire invaluable tools to turn teaching ratio and proportion to professionals and school children into an enjoyable experience. Based on in-depth research, it presents a deep, comprehensive view of the topic, focusing on both the mathematical and psychological-didactical aspects of teaching it. The unique teaching model incorporates both theoretical and practical knowledge, allowing instructors to custom-design teacher courses according to their speci?c needs. The book reports on hands-on experience in the college classes plus teachers' experience in the actual classroom setting. An important feature is the extensive variety of interesting, meaningful authentic activities. While these activities are on a level that will engage pre- and in-service mathematics teachers in training, most can also be utilized in upper elementary and middle school classes. Accompanying the majority of these activities are detailed remarks, explanations, and solutions, along with creative ideas on how to conduct and expand the learning adventure. While primarily written for educators of mathematics teachers, this book can be an invaluable source of information for mathematics teachers of elementary and middle school classes, pre-service teachers, and mathematics education researchers.
Chapters (18)
The concepts of ratio and proportion are fundamental to mathematics and important in many other fields of knowledge. Many phenomena can be expressed as some proportional relationship between specific variables, often leading to some new, unique entity. Conceptualization and comprehension of these concepts, not to mention skills and competence in using them, facilitate mathematic awareness. Even more importantly, these skills foster the ability to use relational reasoning, otherwise known as proportional reasoning, which is crucial to the development of analytical mathematical reasoning.
The information presented in this chapter, combined with the authors’ rationale presented in the introduction, above, provides the conceptual framework of the book, and provides the basis for preparing and organizing courses in ratio and proportion for both pre- and in-service elementary and middle school mathematics teachers: initial training for the former and professional development for the latter.
The model presented below is the basis used for constructing an instructional unit on the topic of ratio and proportion, whether for pre-service mathematics teachers or for in-service mathematics teachers participating in a professional development course.
In mathematics, the concept of ratio is fundamental to many topics. Children encounter the concept in the earliest years of elementary school, even if they are not introduced to the actual word. They first learn the specific word “ratio,” in Grade 6. In fact, many sections of the elementary and middle school curriculum refer to, directly or indirectly, the concept of ratio. Price per item, fractions, percentages, probability, problems in motion, measurement, enlargement and reduction of shapes and figures, and π as a ratio between the circumference of a circle and its diameter, are just a few examples of ratios in the mathematics curriculum of elementary and middle school.
Proportional reasoning is the human ability to make use of an effective form of the proportional scheme. This ability has a central role in the development of mathematical thinking, and is frequently described as a concept that, on the one hand, is a cornerstone of higher mathematics, and, on the other hand, is the peak of the basic tenets of mathematics (Lesh, Post, & Behr, 1988).
Recent studies in many countries worldwide have pointed out the difficulties that pre-service teachers aspiring to teach mathematics in elementary school have. These difficulties are expressed especially as an inability to understand mathematical content and concepts, and feelings of incompetence in dealing with and teaching the subject (Lamon, 2007; Empson & Junk, 2004), combined, among other things, with a certain negative attitude to the mathematics-teaching profession as a whole (Tirosh & Graeber, 1990).
This section presents a wide range of authentic ratio-and-proportion investigative activities that represent actual situations relevant to the real world of students and teachers alike. They span various levels of difficulty appropriate for pre-and inservice elementary and middle school teachers, and can be easily adapted for authentic investigative activities appropriate for pupils in elementary and middle school.
Description: A grade-six teacher wanted to test her students’ previous knowledge and readiness to learn ratio and proportion before she began formal lessons.
Alice and Barbra are teachers. One day they meet at a teacher’s convention. Each has recently purchased a new car, and they begin to compare their automobiles’ features.
Description: Everyone is familiar with the taste-test survey. Here we investigate the results of a cola taste test.
Description: Wimpy visits the Hall of Mirrors in an amusement park and can’t stop laughing. Every mirror shows a different image.
Scientists at a university acquired a small meteoroid that was brought from the moon. When they got to the laboratory to weigh it, they were disappointed to discover that the special scale for weighing small objects was being recalibrated. Anxious to determine the weight of their meteoroid, they searched for another way.
The situations below all have some multiplicative relationship (ratio) between the values. Find the ratio between the values and state if the values have the same unit or if a new unit will be generated (e.g. km/h, individuals per square meter, price per unit, etc.).
The link between assessment and learning is widely acknowledged and highly significant. Sainsbury and Walker (2007), for example, mention four functions of assessment that directly influence student learning: motivating learning, focusing learning, consolidating and structuring learning, and guiding and correcting learning. Rust (2007) concludes that “any scholarship of assessment must therefore be predicated on the value that good assessment supports and positively influences student learning.”
Administration of the attitude questionnaire will give both students and instructors a better understanding and appreciation of the students’ attitude towards ratio and proportion in particular and proportional reasoning in general. Furthermore, administering the attitude questionnaire to student teachers both before and after a pre- or in-service professional development course will serve as an indicator of how the course, by providing instruction to the topics primarily through the use of authentic investigative activities, influenced their attitudes.
Max, Alice, Alex, and Sima planned a class bicycle trip to the zoo. The pupils gathered in the parking lot of the school and rode their bikes along the bike path leading to the zoo. They watched the animals for a few hours, and then met by the lake for some snacks and cold drinks before riding back to the school.
The teaching model presented in this book incorporates into the teaching process the reading and analysis of reports pertaining to both the mathematical and pedagogic-didactic perspectives of ratio and proportion. The guidelines presented above, the results of many years of experience working with pre-service teachers, will guide pre- and in-service teachers in efficiently analyzing such reports.
The annotated articles presented herein are those which were deemed especially appropriate to support, broaden, and deepen the knowledge acquired from performing the authentic investigative activities in the course. Most are from journals specifically geared to elementary and middle school mathematics teachers, and mainly document how various activities in the topic of ratio and proportion were actually applied for use in educational systems and their results.
... Proportional reasoning has been identified as a difficult concept by many educational researchers because it includes a high level of thinking. It requires thinking of more than one quantity, and an understanding the multiplicative relationships between quantities, where this may lead to delay in its development in children (Ben-Chaim et al., 2012;Branch, 2018;Nasir, 2018;Yeong et al., 2020). Many researchers have indicated that teachers at all levels of mathematics education find it difficult to understand proportional reasoning, and they make mathematical errors and misconceptions similar to what students make (Hines and McMahon, 2005;Ruchti, 2005;Lamon, 2007;Glassmeyer et al., 2021;Ozturk et al., 2021). ...
... This study indicated-through a survey of some mathematics teachers and a review of the previous literature on proportional reasoning-that there are multiple difficulties facing teachers and students in the topics related to proportional reasoning. Since many of these difficulties fall into various mathematical errors, there is a primary necessity to overcome these difficulties and address these errors to develop the ability on proportional reasoning, as some of them indicated that mathematics curricula in many countries do not provide opportunities for the teacher to develop proportional reasoning with students (Hines and McMahon, 2005;Ruchti, 2005;Lamon, 2007;Ben-Chaim et al., 2012;Branch, 2018). ...
... After analyzing the unit of proportion prescribed for seventhgrade students and reviewing the previous literature on proportional reasoning (Fernández et al., 2011;Ben-Chaim et al., 2012;Matney et al., 2013;Cruz, 2016;Toluk-Ucar and Bozkus, 2018), the proportional reasoning test was developed according to the basic concepts and skills related to proportional reasoning, which consisted in its initial form of 14 items, each item consisted of the following three tasks: Choose the answer from three choices; give reason of the choice; solve in a way differs from the reasoning in the second task. The test was presented to a group of specialists in mathematics, and mathematics educators. ...
This study investigated the effect of mathematical error analysis-based learning on proportional reasoning ability of seventh-grade students. To achieve the purpose of the study, a proportion unit for the seventh-grade students in Jordan was designed according to the error analysis-based learning. A sample of 45 seventh-grade students participated in the study and were randomly assigned into the following two groups: Experimental group and control group. The data were collected through the following two instruments: A proportional reasoning test and an interview, after ensuring their reliability and validity. The results of the study revealed that the error analysis-based learning led to a significant improvement in proportional reasoning among the experimental group and contributed to providing students with positive experiences in learning mathematics. In light of these results, a set of recommendations for educational researchers, mathematics curriculum designers, and mathematics teachers were presented.
... Además, reflexionar sobre la idoneidad del proceso de instrucción planificado en una lección de libro de texto puede ayudar a los futuros docentes a tomar conciencia sobre sus conocimientos matemáticos y didácticos del razonamiento proporcional y desarrollar aquellos aspectos en los que encuentran mayores dificultades (BEN-CHAIM;KERET;ILANY, 2012;BERK et al., 2009;BUFORN;FERNÁNDEZ, 2018;VAN DOOREN et al., 2008). ...
... Además, reflexionar sobre la idoneidad del proceso de instrucción planificado en una lección de libro de texto puede ayudar a los futuros docentes a tomar conciencia sobre sus conocimientos matemáticos y didácticos del razonamiento proporcional y desarrollar aquellos aspectos en los que encuentran mayores dificultades (BEN-CHAIM;KERET;ILANY, 2012;BERK et al., 2009;BUFORN;FERNÁNDEZ, 2018;VAN DOOREN et al., 2008). ...
... Además, reflexionar sobre la idoneidad del proceso de instrucción planificado en una lección de libro de texto puede ayudar a los futuros docentes a tomar conciencia sobre sus conocimientos matemáticos y didácticos del razonamiento proporcional y desarrollar aquellos aspectos en los que encuentran mayores dificultades (BEN-CHAIM;KERET;ILANY, 2012;BERK et al., 2009;BUFORN;FERNÁNDEZ, 2018;VAN DOOREN et al., 2008). ...
Resumen Una lección de libro de texto describe el proceso instruccional previsto para el estudio de un contenido, por lo que constituye un recurso relevante para los docentes, que deberán ser críticos cuando decidan cómo emplearla en su planificación curricular. En este estudio se describen y analizan las reflexiones que hacen futuros maestros sobre el grado de adecuación de una lección de libro de texto de proporcionalidad, su modo de uso y los cambios que llevarían a cabo para incrementar la idoneidad didáctica del proceso instruccional implementado. El análisis cualitativo de sus informes escritos permite identificar las referencias a criterios de idoneidad didáctica que incluyen en sus valoraciones. Los resultados reflejan que los futuros maestros reflexionan correctamente sobre aspectos epistémicos (falta de argumentación, falta de claridad en la presentación de conceptos, poca variedad de situaciones y representaciones), cognitivos (falta de atención a conocimientos previos, no advertencia de errores y dificultades al alumno) e instruccionales (la lección prioriza el aspecto procedimental y deja interacciones entre alumnos en segundo plano). Aunque la mayoría identifica conflictos semióticos en la lección y reconoce que el texto debe ser un recurso sobre el cual ha de realizarse cambios para una gestión eficiente, las modificaciones que proponen refieren sólo de modo parcial a la valoración realizada previamente, priorizando cambios en lo epistémico (variar tipología de tareas) y cognitivo (incluir resúmenes de contenidos previos). Concluimos este estudio con una propuesta de mejoras para posteriores intervenciones formativas.
... A missing value problem is one in which three of the four values in a proportion are given, with the purpose of finding the missing value (Lamon, 2007). While solving missing value problems, which is one of the most preferred proportional problem types in mathematics lessons, it is seen that the students use the crossmultiplication algorithm without noticing the multiplicative relations between the quantities (Ben-Chaim et al., 2012;Duatepe et al., 2005;Lamon, 2007). In numerical comparison problems, all four quantities that make up two ratios are given, and the goal is to determine if they are equal or if one ratio is higher or smaller than the other (Ben-Chaim et al., 2012). ...
... While solving missing value problems, which is one of the most preferred proportional problem types in mathematics lessons, it is seen that the students use the crossmultiplication algorithm without noticing the multiplicative relations between the quantities (Ben-Chaim et al., 2012;Duatepe et al., 2005;Lamon, 2007). In numerical comparison problems, all four quantities that make up two ratios are given, and the goal is to determine if they are equal or if one ratio is higher or smaller than the other (Ben-Chaim et al., 2012). In this problem type, students were more likely to use the unit rate technique (Duatepe et al., 2005;Pakmak, 2014). ...
... Recognizing and solving additive relationships in non-proportional problems could be accomplished by going beyond rote-based methods. Therefore, in ratio and proportion teaching, additive relations between variables in nonproportional problems should be included, as well as multiplicative relations between proportional variables (Ben-Chaim et al., 2012;Karplus et al., 1983). ...
The aim of this study was to examine the proportional reasoning skills of seventh-grade students before and after the implementation of STEM activities involving proportional and non-proportional relationships. Case study, one of the qualitative study methods, was used in the research. The data for the study was obtained from eight students. Seven different STEM activities were implemented over a seven-week period. A Proportional Reasoning Test and semi-structured interviews were used as data collection tools before and after the implementation of STEM activities. The findings revealed that the STEM activities contributed to the development of students’ proportional reasoning skills. Before the STEM activities, students mostly used a cross-multiplication strategy to solve proportional problems. Moreover, they had difficulties in solving numerical comparison and qualitative reasoning problems. Additionally, students frequently used multiplicative relations in non-proportional problems. After the STEM activities, students used multiplicative relations to solve proportional problems. Furthermore, students could solve numerical comparison and qualitative reasoning problems.
... It also provides a mathematical basis for more complex concepts in high school (Lamon, 2012). However, research showed that students' proportional reasoning abilities were generally problematic (Ayan & Işıksal-Bostan, 2018, 2019Atabaş & Öner, 2016;Ben-Chaim, Keret, & Ilany, 2012;Behr, Lesh, Post, & Silver, 1983;Cramer, Post, & Behr, 1989;Hart, 1988;Lamon, 2007;Özgün-Koca & Kayhan-Altay, 2009;Toluk-Uçar & Bozkuş, 2018). First of all, students frequently used limited number of strategies and mostly formal strategies, which do not highlight multiplicative relationships, to set up and solve proportional problems (Ayan & Işıksal-Bostan, 2019;Ben-Chaim et al., 2012;Özgün-Koca & Kayhan-Altay, 2009;Toluk-Uçar & Bozkuş, 2018). ...
... However, research showed that students' proportional reasoning abilities were generally problematic (Ayan & Işıksal-Bostan, 2018, 2019Atabaş & Öner, 2016;Ben-Chaim, Keret, & Ilany, 2012;Behr, Lesh, Post, & Silver, 1983;Cramer, Post, & Behr, 1989;Hart, 1988;Lamon, 2007;Özgün-Koca & Kayhan-Altay, 2009;Toluk-Uçar & Bozkuş, 2018). First of all, students frequently used limited number of strategies and mostly formal strategies, which do not highlight multiplicative relationships, to set up and solve proportional problems (Ayan & Işıksal-Bostan, 2019;Ben-Chaim et al., 2012;Özgün-Koca & Kayhan-Altay, 2009;Toluk-Uçar & Bozkuş, 2018). Moreover, students generally had difficulties in distinguishing proportional from non-proportional situations (Ayan & Işıksal-Bostan, 2018, 2019Atabaş & Öner, 2016;Toluk-Uçar & Bozkuş, 2018). ...
... Additionally, in a numerical comparison problem, all of the four values that form two ratios (a, b, c, and d) are provided and the aim is "to determine the order relation between the ratios ⁄ ⁄ " (Lamon, 2007, p. 637). In other words, the numerical comparison problems require the comparison of two ratios in order to determine whether the two ratios are equal or which ratio is greater or smaller than the other one (Ben-Chaim et al., 2012). Moreover, qualitative reasoning problems do not include numerical values; however, they "require the counterbalancing of variables in measure spaces" (Cramer et al., 1993, p. 166). ...
... Furthermore, various pieces of research show that teachers, both in their initial training and in service, show difficulties with teaching concepts relating to proportionality (Bartell, Webel, Bowen, & Dyson, 2013;Ben-Chaim, Keret, & Ilany, 2012;Berk, Taber, Gorowara, & Poetzl, 2009;. Teacher training must take into account the development of mathematics teaching knowledge and competences relating to this topic by designing and implementing specific training interventions. ...
... The educational video analysed 1 covers the topic of direct arithmetic proportionality from the perspective of presenting the notions of ratio and proportion (Ben-Chaim, Keret, & Ilany, 2012). The idea is that the future teachers will watch the video closely and critically assess its degree of suitability, in line with the epistemic suitability components and indicators Godino, 2013). ...
... Consideramos que sería deseable que los profesores conozcan la herramienta idoneidad didáctica y adquieran competencia para su uso en el análisis crítico de los recursos educativos, particularmente del uso de vídeos disponibles en Internet. Por otra parte, diversas investigaciones señalan que tanto los profesores en formación inicial como en servicio presentan dificultades para enseñar conceptos relacionados con la proporcionalidad (Bartell, Webel, Bowen y Dyson, 2013;Ben-Chaim, Keret e Ilany, 2012;Berk, Taber, Gorowara y Poetzl, 2009;Hilton y Hilton, 2018). Se hace preciso que la formación de profesores tenga en cuenta el desarrollo de conocimientos y competencias didáctico-matemáticas con relación a este tema, diseñando e implementado intervenciones formativas específicas. ...
... Regarding the importance of investigating learning obstacle in mathematics teaching and learning, one of the topics that shows the existence of learning obstacle is ratio and proportion. This topic is listed in curriculum in Indonesia [11] and becomes a cornerstone for understanding the other mathematical topics [12,13]. In addition, the students' understanding will lead them to develop mathematical skills [14] and help them to solve real-life problems [15,16], consequently they have to master it. ...
... where 0 b [13]. It indicates that the understanding of fraction concept is required. ...
... The students also have to know about distinguishing whether a condition is ratio, and the concept involved. If they understand the fraction concept and its application, they will realize a condition where zero is only for numerator [13], as the example. Those are the aspects why they do not achieve the understanding of ratio and proportion concept. ...
Many studies reported that the students lack understanding on the topic of ratio and proportion. This topic basically becomes a cornerstone for understanding the other mathematics topics, developing mathematical skills, and helping the students to solve real-life problems, but many researchers have documented the obstacles regarding this topic. For such a reason, this study intends to investigate kinds of learning obstacles on the topic of ratio and proportion that can emerge due to some causes. It was part of didactical design research which was conducted to eighth graders who had already been taught about its topic. The data were collected from students’ answer and interview in solving ratio and proportion problems. The data analysis reveals that the obstacles are classified into ontogenic obstacle, didactical obstacle, and epistemological obstacle. It can be identified from students’ understanding or conception about ratio and proportion which was caused by students’ prior knowledge, didactical practices, or the lack of contexts. These findings are expected to overcome learning obstacles in teaching and learning of ratio and proportion.
... Proportional reasoning is a unifying concept which integrates many mathematical topics in middle school [1] and a key theme in an extensive range of essential topics beyond grades 5-8 [2]. Mathematics-related topics for which proportional reasoning is essential include similarity, scaled drawings [3][4][5], problem solving and measurement [5,6]. In addition, it lies at the heart of many topics in science and situations in daily life [5,7]. ...
... Proportional reasoning is a unifying concept which integrates many mathematical topics in middle school [1] and a key theme in an extensive range of essential topics beyond grades 5-8 [2]. Mathematics-related topics for which proportional reasoning is essential include similarity, scaled drawings [3][4][5], problem solving and measurement [5,6]. In addition, it lies at the heart of many topics in science and situations in daily life [5,7]. ...
... Mathematics-related topics for which proportional reasoning is essential include similarity, scaled drawings [3][4][5], problem solving and measurement [5,6]. In addition, it lies at the heart of many topics in science and situations in daily life [5,7]. It is no surprise, then, that proportional reasoning is referred to as a watershed concept, a cornerstone of elementary mathematical concepts [8] and a gateway to higher mathematics and mathematical success [9]. ...
The purpose of this study was to examine middle school students’ proportional reasoning, solution strategies and difficulties in real life contexts in the domain of geometry and measurement. The underlying reasons of the difficulties were investigated as well. Mixed research design was adopted for the aims of the study by collecting data through an achievement test from 935 sixth, seventh and eighth grade students. The achievement test included real life problems that required proportional reasoning, and were related to the measurement of length, perimeter, area and volume concepts. In addition, task-based interviews were conducted on 12 of these students to collect more comprehensive data and to support the findings of the achievement test. Findings revealed that although students were mostly successful in giving correct answers, their reasoning lacked a clear argument of the direct and indirect proportional relationships between the variables and that they approached the problems by superficial characteristics of the problems.
... Sin embargo, la proporcionalidad no suele recibir un tratamiento adecuado en los textos de matemáticas de ambas etapas, observándose un predominante aspecto algorítmico que dificulta el desarrollo de un adecuado razonamiento proporcional en los escolares (AHL, 2016;. Además, tanto profesores en formación como en servicio tienen dificultades para enseñar conceptos relacionados con la proporcionalidad (BEN-CHAIM; KERET;ILANY, 2012;BUFORN;LLINARES;FERNÁNDEZ, 2018;WEILAND et al., 2021). En este sentido, autores como Remillard y Kim (2017) sugieren que es posible diagnosticar y corregir estas deficiencias por medio de la reflexión sobre los procesos instruccionales previstos en lecciones de libros de texto, generando aprendizaje significativo en los docentes, lo que justifica el interés de nuestra propuesta. ...
... Sin embargo, la proporcionalidad no suele recibir un tratamiento adecuado en los textos de matemáticas de ambas etapas, observándose un predominante aspecto algorítmico que dificulta el desarrollo de un adecuado razonamiento proporcional en los escolares (AHL, 2016;. Además, tanto profesores en formación como en servicio tienen dificultades para enseñar conceptos relacionados con la proporcionalidad (BEN-CHAIM; KERET;ILANY, 2012;BUFORN;LLINARES;FERNÁNDEZ, 2018;WEILAND et al., 2021). En este sentido, autores como Remillard y Kim (2017) sugieren que es posible diagnosticar y corregir estas deficiencias por medio de la reflexión sobre los procesos instruccionales previstos en lecciones de libros de texto, generando aprendizaje significativo en los docentes, lo que justifica el interés de nuestra propuesta. ...
Una lección de un libro de texto muestra el proceso de instrucción planificado por el autor como medio para favorecer el aprendizaje de un contenido por parte de potenciales estudiantes. Por tanto, es esencial que el profesor que decide usar una determinada lección en sus clases analice y valore lo que ocurre en dicho proceso. Determinar la adecuación de una lección precisa de un primer análisis detallado de las situaciones de enseñanza, describiendo la secuencia de prácticas operativas y discursivas que propone el autor y caracterizando los objetos y procesos como elementos constitutivos del contenido matemático que debe ser aprendido. Este análisis permitirá atender a cómo se gestionan los conocimientos previos requeridos e identificar su papel en las posibles dificultades de aprendizaje. A continuación, la valoración de la idoneidad didáctica del proceso instruccional basado en el uso de la lección orienta al profesor en la toma de decisiones sobre la gestión del recurso. En este trabajo se describe, por medio de un estudio de caso, una experiencia formativa con maestros en formación destinada a desarrollar la competencia de análisis didáctico de lecciones de libros de texto. El diseño, implementación y evaluación de la experiencia, están basados en la aplicación de herramientas teórico-metodológicas del Enfoque Ontosemiótico, en particular las nociones de significado pragmático y la de configuración ontosemiótica de prácticas, objetos y procesos para el análisis de la actividad matemática, y la teoría de la idoneidad didáctica, en la valoración de la adecuación de los procesos de enseñanza y aprendizaje.
... Con frecuencia los docentes relegan el desarrollo de una comprensión conceptual, centrando la atención en el aspecto operacional y justificando sus estrategias de resolución en problemas de proporcionalidad en argumentos procedimentales (Lamon, 2007;Riley, 2010;Post, Harel, Behr y Lesh, 1991). Ben-Chaim et al. (2012) recurren en su investigación a tareas matemáticas para promover el conocimiento didáctico-matemático de futuros profesores en relación al razonamiento proporcional, planteando que los futuros profesores lean artículos didáctico-matemáticos sobre razón y proporción y contemplando el trabajo y discusión de resultados en grupo. Por otro lado, Berk y cols. ...
... Tareas propuestas a los alumnos (inspiradas enBen-Chaim et al.;2012) ...
En este trabajo se describe el diseño, implementación y análisis retrospectivo de una acción formativa con estudiantes del grado de Educación Primaria, cuyo objetivo es evaluar el conocimiento sobre proporcionalidad y el grado de desarrollo de dos aspectos relevantes del conocimiento didáctico-matemático de dicho contenido: el análisis de objetos y significados puestos en juego en las prácticas matemáticas y el estudio de niveles de algebrización involucrados en distintas soluciones a problemas de proporcionalidad. La experiencia formativa se ha realizado con un grupo de 35 estudiantes en el marco de una asignatura sobre diseño y desarrollo del currículo en Educación Primaria en la cual se atribuye un papel relevante al trabajo en equipo. Los resultados indican que los conocimientos y competencias especializadas de los estudiantes sobre proporcionalidad presentan lagunas específicas que pueden dificultar la enseñanza del tema. Los estudiantes han logrado competencia para identificar el sistema de prácticas elementales en la resolución de las tareas y reconocer los niveles de algebrización puestos en juego. Pero se requiere mayor tiempo para que los futuros profesores sean capaces de identificar los distintos objetos que intervienen en las prácticas matemáticas y para enunciar variantes pertinentes de un problema dado.
... Proportional reasoning is a mathematical content of special interest in this line of research because many topics within the school mathematics curriculum (scale, probability, percent, rate, trigonometry, equivalence, measurement, algebra, the geometry of plane shapes) and science (density, molarity, speed and acceleration, force) require knowledge and understanding of ratio and proportion (Dole et al. 2012). However, several investigations indicate that both prospective and in-service teachers have difficulties in teaching concepts related to proportionality (Ben-Chaim et al. 2012;Berk et al. 2009;Buforn and Fernández 2014;Hilton and Hilton 2018;Lamon 2007;Livy and Vale 2011;Riley 2010;Aké et al. 2013;Burgos et al. 2018;Font et al. 2013;Giacomone et al. 2019;Godino et al. 2007Godino et al. , 2013Godino et al. , 2014Godino et al. , 2017. Consequently, teachers' education should develop mathematical and didactical knowledge and competence regarding proportionality, through specific formative activities. ...
... Despite the importance of this content, many investigations reveal that both prospective and in-service teachers have difficulties in understanding and teaching some of the proportional reasoning components (Ben-Chaim et al. 2012;Berk et al. 2009;Buforn et al. 2018;Livy and Vale 2011;Riley 2010). For example, Son (2013) investigated elementary and secondary pre-service teachers' reasoning, their responses to student errors on the topic of ratio and proportion, and the relationship between their knowledge and their pedagogic approaches. ...
In this paper, we describe the design, implementation and results of a formative intervention, meant to develop prospective primary school teachers’ competence in the analysis of the difficulties emerging in the resolution of proportionality tasks. Solving problems by different methods, identifying the knowledge put at stake in each problem and taking this information into account to recognize the difficulties that pupils may encounter in solving the problems using each strategy are essential aspects of the epistemic and cognitive facets of didactic-mathematical knowledge. The experience has been carried out with a sample of 88 prospective primary school teachers during the third year of their studies, by applying a didactic model that includes work in teams, institutionalization, and assessment of the individual learning achieved. To analyse the participants’ responses, we used some theoretical and methodological tools of the onto-semiotic approach in mathematics education. The identification of pupils’ potential difficulties in addressing problem-solving, based on the objects involved in the mathematical activity, was a challenging task for prospective teachers. The difficulties most frequently identified by the prospective teachers were those concerned with understanding the statement requirements, or the problem-situations context, and the mathematical procedures involved in the resolution of the task. They did not discriminate the difficulties according to the resolution strategies. Time constraints conditioned the degree of learning achieved. We conclude that it is necessary that teacher education programs consider this type of didactical analysis and should be articulated with situations focused on developing other complementary knowledge and competencies.
... where m is non zero (Ben-Chaim, Keret, Ilany, 2012). An example of direct proportion is the result of the distance traveled by car (s) over time (t), which is the velocity (v). ...
... or p q r s where m is non zero (Ben-Chaim, Keret, Ilany, 2012). An example of inverse proportion is the relationship between the speed of a car (v) and its travel time (t), which means the distance (s) is constant. ...
... En esta solución (semejante a la de "división por la razón" en Ben-Chaim et al., 2012) intervienen valores numéricos particulares y se aplican operaciones aritméticas sobre dichos valores. La igualdad tiene significado de resultado de una operación. ...
Diversas investigaciones señalan las carencias en el razonamiento probabilístico de futuros docentes y su conexión con un razonamiento proporcional deficiente. Estas limitaciones pueden estar relacionadas además con el grado de algebrización de la actividad matemática implicada. Con la intención de arrojar algo de luz al respecto, en este estudio, se analizan las respuestas de un grupo de maestros en formación a una tarea que requiere determinar la composición de una urna, cuya probabilidad de éxito es igual que en otra en la que se conoce la razón entre casos favorables y desfavorables. Se examinan las estrategias y errores que presentan, centrándonos en los niveles de razonamiento algebraico de sus prácticas matemáticas. Los resultados muestran que los futuros docentes determinaron con éxito la composición de la urna empleando estrategias mayoritariamente de tipo aritmético, y que encontraron dificultades para argumentar sus soluciones. Estas dificultades fueron menores a medida que las soluciones mostraban rasgos de razonamiento proto-algebraico.
... More simply, this proportional relationship can be represented as a linear function y=kx for equal proportions or = 1 for inverse proportions (Lamon, 2020). In addition to being represented as a linear function, proportional problems can also be interpreted as problems involving two equivalent ratios (Ben-Chaim et al., 2012). Ratio is a term that refers to the comparison of two or more quantities (Lamon, 2020). ...
Problem solving for proportion problems has not fully incorporated various appropriate strategies. This study aimed to identify and analyze the strategy used by pre-service mathematics teachers in solving proportion problems. This research used a qualitative method with a phenomenological design. The participants of this study were 29 pre-service mathematics teachers with the characteristics of having learned the concept of proportion. Data were collected using tests, interviews, observation, and document study techniques. Data were analyzed in stages, starting with data collection, data reduction, data review, and results conclusion, to solve proportion problems. The results obtained were in the form of a description of the techniques used in solving proportion tasks. Pre-service mathematics teachers mostly used the cross-product strategy in solving proportion tasks. The results of this study can be used as a basis for developing hypothetical learning trajectories for comparison learning for pre-service mathematics teachers in the future. Abstrak Penyelesaian soal proporsi belum sepenuhnya memasukkan berbagai strategi yang tepat. Penelitian ini bertujuan untuk mengidentifikasi dan menganalisis strategi yang digunakan guru matematika calon guru dalam menyelesaikan masalah perbandingan. Penelitian ini menggunakan metode kualitatif dengan desain fenomenologis. Partisipan penelitian ini berjumlah 29 mahasiswa calon guru matematika dengan karakteristik telah mempelajari konsep proporsi. Data dikumpulkan dengan menggunakan teknik tes, wawancara, observasi, dan studi dokumen. Data dianalisis secara bertahap, dimulai dari pengumpulan data, reduksi data, penelaahan data, dan penarikan kesimpulan hasil, untuk menyelesaikan masalah proporsi. Hasil yang diperoleh berupa uraian tentang teknik-teknik yang digunakan dalam menyelesaikan tugas proporsi. Mahasiswa calon guru matematika sebagian besar menggunakan strategi cross produk dalam menyelesaikan tugas proporsi. Hasil penelitian ini dapat digunakan sebagai dasar untuk mengembangkan hypothetical learning trajectory untuk pembelajaran perbandingan bagi mahasiswa calon guru matematika di masa yang akan datang.
... The wide research on proportional reasoning has focused on the different meanings of the rational number (Burgos & Godino, 2020). This research is compiled, among other papers, by Ben-Chaim et al. (2012), Carpenter et al. (2012), Lamon (2007), Kieren (2020), Obando et al. (2014), andVan Dooren et al. (2018). ...
This research aimed to relate Costa Rican students (11-16-year-olds) competence to compare probabilities in spinners and proportional reasoning in the comparison of ratios. We gave one of two questionnaires to a sample of 292 students (grade 6 to grade 10) with three probability comparison and three ratio comparison problems each. Globally both questionnaires cover six different proportional reasoning levels for each type of problem. Additionally, each questionnaire contains two comparison probabilities items intended to discover a specific bias. We analyze the percentages of correct responses to the items, strategies used to compare probabilities per school grade, and students’ probabilistic reasoning level. The results confirm more difficulty in comparing ratio than in comparing probability and suggest that the reasoning level achieved is lower than established in previous research. The main bias in the students’ responses was to consider the physical distribution of colored sectors in the spinners. Equiprobability and outcome approach were very scarce.
... Conscientes de que tanto los profesores en formación inicial como en servicio presentan dificultades para enseñar conceptos relacionados con la proporcionalidad (Bartell et al., 2013;Ben-Chaim et al., 2012;Berk et al., 2009), Burgos et al. (2020) llevaron a cabo una acción formativa con futuros maestros de educación primaria, orientada al desarrollo de la competencia de análisis de la idoneidad didáctica de vídeos sobre proporcionalidad. En dicha investigación el instrumento de evaluación usado se centró únicamente en el contenido matemático. ...
This paper describes the design, implementation and results of a formative action with 61 students for primary education teachers, oriented to the development of the competence of analysis of the didactic suitability of educational videos about percentages. The a priori analysis of the video made by the researchers revealed important deficiencies in the different facets of didactic suitability, being especially relevant those detected in the epistemic dimension. Most of the students for teacher made quite relevant global assessments of the didactic suitability (70.49% considered it between low and medium) basing their judgment on the precision or degree of fulfillment of the suitability indicators in the different components. However, although the suggestions for improvement indicated are quite timely, not all students who identify weaknesses in the videos are able to explicitly make changes to the resource.
... Student textbooks emphasise rote learning and avoid arguing about the conditions that characterise a situation of direct proportionality, which hinders the development of adequate proportional reasoning (Fernández & Llinares, 2011;Lamon, 2007;Riley, 2010). Furthermore, and not least, both pre-service and in-service teachers have difficulties in teaching concepts related to proportionality (Ben-Chaim et al., 2012;Berk et al., 2009;Buforn et al., 2018;Van Dooren et al., 2008). In this sense, authors such as Nicol and Crespo (2006) or Remillard and Kim (2017) suggest that it is possible to diagnose and correct these deficiencies by reflecting on the instructional processes envisaged in textbook lessons, generating meaningful learning in teachers. ...
Background: Being able to assess what is happening in a teaching-learning process is one of the teacher's competencies. Teachers often must analyse and select the educational resources they consider relevant for their students. Since textbooks are an important tool for instructional design, the teacher must be able to analyse their suitability, identify limitations and make adaptations to overcome them. Objectives: This paper describes the design, implementation, and results of a training action with prospective primary education teachers, aimed at the development of competence for the analysis, identification of conflicts and proposals for the management of textbook lessons, particularly the content of proportionality. Design: The research is interpretative and exploratory, using content analysis to examine the participants' response protocols. Setting and Participants: The experience was carried out in the framework of the Primary Education grade; the sample consisted of 48 students. Data collection and analysis: Data were collected by observer/researcher annotations and participants' responses to the proposed assessment task. Results: The results show that trainee teachers make progress in identifying conflicts and make suggestions for improvement to increase didactic suitability in the different facets, but that they are not specific enough when describing effective conflict resolution proposals, especially in the epistemic and cognitive facets. Conclusions: For future teachers to become more proficient in the mode of use, it is necessary to reinforce their didactical-mathematical knowledge of proportionality.
... The teaching and usage of proportional reasoning in everyday life will become more difficult if it is not understood conceptually but algorithmically (Dooley, 2006). Not only experienced by elementary and middle students, but classroom teachers also have difficulty understanding proportional relationships (Ben-Chaim et al., 2012;Irfan et al., 2018;Jacobson & Izsák, 2014;Lamon, 2007). ...
p style="text-align: justify;">Teacher knowledge is one of the main factors in the quality of mathematics learning. Many mathematics teachers have difficulty using proportional reasoning. Proportional reasoning is one of the essential aspects of the middle school mathematics curriculum to develop students' mathematical thinking. Teachers should realize that developing proportional reasoning is not an easy task. In this study, we investigated how teachers give proportional reasoning about the concept of proportional and non-proportional situations, especially in making sense of them. The research subjects were mathematics teachers who had taught proportional-related material. Data was collected using task-based interviews outside the teacher's working hours. Data analysis and interpretation were completed using a framework meaning-based approach. The results of the data analysis showed that the teacher is careful in understanding information, is aware of multiple meanings, and knows key information in understanding the contextual structure of proportional and non-proportional situations. Furthermore, they are also able to identify additive and multiplication relationships, have flexibility in understanding proportional and non-proportional situations separately or collectively, and understand problem-solving systematics in detail.</p
... La marcada influencia en la mayoría de los textos del aprendizaje memorístico de rutinas obstaculiza el desarrollo de un adecuado razonamiento proporcional (Fernández y Llinares, 2011;Lamon, 2007;Riley, 2010). A esto se suma que tanto los profesores en formación inicial como en servicio muestran una comprensión limitada de los significados de razón y proporción, así como dificultades para enseñar conceptos relacionados con la proporcionalidad (Ben-Chaim et al., 2012;Berk et al., 2009;Buforn et al., 2018;Van Dooren et al., 2008). ...
Se describe el diseño, implementación y resultados de una experiencia formativa con futuros maestros destinada a fomentar su competencia reflexiva sobre procesos de enseñanza y aprendizaje. Se propone a los participantes analizar una lección de proporcionalidad, empleando una guía basada en indicadores de idoneidad didáctica según el Enfoque Ontosemiótico. La evaluación de los análisis elaborados por los futuros maestros permite detectar carencias en conocimientos matemáticos y didácticos sobre la proporcionalidad que impiden una interpretación adecuada de los indicadores y una correcta valoración de estos sobre la lección. Planteamos estrategias a considerar en nuevas intervenciones formativas para superar estas dificultades.
... A pesar de su importancia, numerosas investigaciones revelan que tanto profesores en formación como en ejercicio presentan dificultades para comprender y enseñar algunos de los componentes que constituyen el razonamiento proporcional (Ben-Chaim et al., 2012;Buforn et al., 2018;Burgos y Godino, 2022a;Izsák y Jacobson, 2017;Hilton y Hilton, 2019;Nagar et al., 2016;Weiland et al., 2019), así como para interpretar el pensamiento matemático de los alumnos cuando resuelven problemas de proporcionalidad (Buforn et al., 2020;Burgos y Godino, 2022B;Fernández et al., 2012Fernández et al., , 2013. ...
Para garantizar un proceso de enseñanza y aprendizaje óptimo, el profesor de matemáticas debe ser capaz de analizar, interpretar y valorar la actividad matemática que desarrollan sus alumnos cuando resuelven las tareas que les propone. Esta competencia permite al profesor comprender los logros de aprendizaje y las dificultades que muestran los estudiantes de cara a tomar decisiones de acción pertinentes. El objetivo de este trabajo es describir y analizar la competencia de un grupo de 130 futuros maestros de educación primaria para interpretar las respuestas de estudiantes ante una situación de proporcionalidad (problema de comparación). Entre los resultados obtenidos destacamos un conocimiento didáctico-matemático del razonamiento proporcional insuficiente que impide a los futuros maestros interpretar de forma pertinente el grado de corrección en las soluciones o el pensamiento matemático aparente de los estudiantes.
... From an epistemic perspective, i.e., from institutionalised mathematical knowledge, proportionality has been studied fundamentally based on three approaches: arithmetic, focused on the notions of ratio and proportion (where problems of comparison or missing value stand out); the algebraic-functional, based on the notion and properties of the linear function; and the geometric, focused on the similarity of figures. Although the first of these meanings predominates in most curricular proposals and research (Ben-Chaim et al., 2012;Lamon, 2007), many authors defend the importance of beginning the study of proportionality with an informal approach, before formalising the concepts of ratio and proportion (Cramer & Post, 1993;Ruiz & Valdemoros, 2004). Based on perceptive comparison and qualitative analysis of the multiplicative relationships between particular numbers, this intuitivetype perspective is proposed as the first approach to proportionality (Burgos & Godino, 2020a;Fiol & Fortuny, 1990). ...
Background: Problem posing is a fundamental competence that enhances the didactic-mathematical knowledge of the mathematics teacher, so it should be an objective in teacher education plans. Objectives : This paper describes and analyses a training intervention with prospective teachers to develop such competence using proportionality tasks. Design: This qualitative and interpretative study adopts a methodology characteristic of didactic design or engineering research. The design of the intervention and the content analysis of the participants’ answers use theoretical and methodological tools from the onto-semiotic approach to mathematical knowledge and instruction. Context and participants : The training action was carried out with 127 undergraduates attending a primary education teaching degree in the framework of the Design and Development of the Mathematics Curriculum in Primary Education subject in a Spanish university. Data collection and analysis: The prospective teachers, organised in 33 working teams, were asked to create two problems based on a given situation and to identify objects and difficulties, the solution of which was analysed by the research team. Results: The results show that the participants encounter difficulties in elaborating relevant proportionality problems from the given situation, identifying the associated level of complexity, recognising the mathematical objects interacting in the solution to their problems and the difficulties that these could cause to primary school pupils. Conclusions: It is mandatory to reinforce problem creation competence and proportional reasoning in teacher education.
... If you look at it starting from the elementary school level to senior high school, in the case the ratio and proportion material is a material that must be taught on an ongoing basis (Kemendikbud, 2016b). In mathematics, the concepts of ratios and proportions are foundational material for other sciences due to the fact many parts of the primary and secondary school curriculum draw on the concepts of ratios and proportions, such as item values, fractions, percentages, probabilities, measurements, transformations of shapes and figures (Ben-Chaim et al., 2012). Johar et al., (2018) Several textbooks and inverse proportion studies were also carried out. ...
Buku teks yang memiliki konten isi memiliki peranan yang sangat penting dalam menunjang proses pengetahuan siswa secara utuh. Fakta menunjukkan bahwa sebagian besar guru bergantung pada buku teks, hingga ada keluhan pada isi dan sajian buku teks. Penelitian ini bertujuan untuk mendeskripsikan karakteristik sajian buku teks kelas tujuh pada materi perbandingan berbalik nilai ditinjau dari teori prakseologi yang memiliki empat elemen analisis, yaitu tugas, teknik, teknologi, teori. Penelitian ini adalah penelitian deskriptif kualitatif dengan objek penelitian adalah buku teks siswa kelas VII terbitan Kemendikbud, serta subjek penelitian melibatkan guru kelas tujuh. Temuan penelitian menunjukkan bahwa sajian buku teks materi perbandingan berbalik nilai sudah cukup baik. Namun, ada beberapa sajian yang perlu dilengkapi. Materi perbandingan berbalik nilai sudah bersifat kontekstual, tetapi terdapat ilustrasi tidak sesuai perkembangan kognitif siswa. Sajian teknik kurang lengkap dan tidak memberikan ruang bagi siswa untuk memilih cara menyelesaikan kasus perbandingan berbalik nilai, tidak ada kesempatan bagi siswa untuk memberikan justifikasi terhadap teknik yang dilakukan, serta beberapa tidak ada situasi untuk membuat siswa menyimpulkan terhadap hasil justifikasi yang dilakukan. Textbooks with content are very important in supporting the students’ knowledge process. The facts show that most teachers depend on textbooks, so there are complaints about the content and presentation of textbooks. This study aims to describe the characteristics of a seventh-grade textbook presented on inverse proportion in terms of the praxeological theory, which has four elements of analysis, namely task, technique, technology, and theory. This research is a qualitative descriptive study with the object of research being a textbook of class VII students published by the Ministry of Education and Culture. The research subject involves seventh-grade teachers. The study results show that the presentation of inverse proportion in textbooks is good enough. However, there is some present that needs to be completed. The inverse proportion is already contextual, but some illustrations do not match students' cognitive development. The presentation of techniques is incomplete and does not provide space for students to choose how to solve cases of inverse proportion, there is no opportunity for students to justify the techniques used, and in some situations, there are no situations to make students conclude on the results of the justifications carried out.
... El primer indicador en este subcomponente contempla la necesidad de presentar de forma clara los conceptos fundamentales de la proporcionalidad, según el nivel educativo; para el caso de secundaria sería necesario distinguir entre razón, tasa, proporción, porcentaje, fracción y número racional, entre otros (Aroza et al., 2016). Este aspecto recibe especial atención en el enfoque aritmético de la proporcionalidad, que tiene un carácter predominante en varias propuestas curriculares e investigaciones (Ben-Chaim et al., 2012;Fernández y Puig, 2002;Hart, 1998;Lamon, 2007;Shield y Dole, 2013). Para Fernández y Puig (2002), el estatuto lógico de la razón es de un nivel más elevado que el de número, fracción, longitud y otros conceptos previos, ya que proviene de una relación de equivalencia. ...
The objective of this study is the construction of a guideline of indicators of didactic suitability related to the topic of proportionality, which can be used as a basis for the analysis and evaluation of textbook lessons, and as a resource for teachers’ reflection on effectively implemented instructional processes about proportionality. Since the study of proportional reasoning has become a relevant field of research in mathematics education, it is possible to identify specific suitability criteria for guiding the teaching and learning processes, applying the methodology of content analysis to a bibliographic review of key research in this area. Various aspects, components, and indicators of the notion of didactic suitability guide the selection and categorization of didactic-mathematical knowledge about proportionality in elementary and secondary education derived from research about such content. Finally, a Mathematics Textbook Lesson Analysis Guide adapted to the topic of proportionality was created. Since the textbook contains curricular material that largely determines what happens in the classroom and acts as a mediator in student learning, and ratios, proportions, and proportionality are studied in the curricula of elementary and secondary education, the Guide obtained can be a valuable resource for teachers. It is necessary to design and implement training actions with teachers that they are familiar with and can use competently, bearing in mind that there is always room for further improvement.
... This study has analysed the effect of authentic investigation activities developed by Ben-Chaim, Ilany & Keret (2012) on the process of reasoning of ratio concept. ...
"Abstract: The aim of this research is to examine the effect of authentic investigation activities on the reasoning process of ratio concept which has been regarded as the milestone of mathematics and science teaching. In line with this purpose, the research has been carried out with eight 7th grade students attending a public school located in Central Anatolia Region in Turkey. The success levels of these students in mathematics are in a heterogeneous structure ranging from higher and moderate to lower levels. The data of this qualitative study has been collected through authentic investigation activities, researcher logs, and interviews and has been analysed via descriptive techniques. At the end of the study, it has been concluded that the teaching environment containing authentic investigation activities has a positive effect on the reasoning process of students on ratio concept."
... En el ámbito de la enseñanza superior y específicamente en la formación de profesores, se observa un mayor interés en el desarrollo de investigaciones relativas al estudio del conocimiento matemático y didáctico considerado necesario para enseñar la proporcionalidad (Lo, 2004;Rivas, Godino y Castro, 2012). Existen evidencias empíricas que el razonamiento proporcional sigue M siendo problemático para muchos estudiantes universitarios (Harries y Botha, 2013;Frith y Lloyd, 2016;Ben-Chaim, Ilany y Keret, 2012). Otros estudios corroboran los resultados de que estudiantes de pregrado e incluso docentes, tienen problemas en los temas de razón, proporción y fracciones reductoras (Çalışıcı, 2018;Bingölbali y Özmantar, 2010;Kaplan, İşleyen y Öztürk, 2011;Ekawati, Lin y Yang, 2018). ...
Este artículo tiene como objetivos determinar y analizar el desempeño de los futuros profesores de matemática de enseñanza secundaria en la resolución de tipos de problemas y los niveles del razonamiento proporcional. La investigación es descriptiva con metodología cuantitativa con una muestra de veinticinco estudiantes de una Universidad en Chile. Los datos se recopilan a través de una prueba de problemas de respuesta abierta sobre aplicaciones de la proporcionalidad. Los resultados develan capacidad de los estudiantes para resolver problemas rutinarios, preferentemente de contexto puramente matemático y en menor grado, fantasista, pero con alta dificultad en la resolución de problemas no rutinarios y de contexto real. La categoría de nivel aditivo es la mayormente utilizada, lo que demuestra la prevalencia del razonamiento pre-proporcional sin lograr alcanzar la categoría de nivel proporcional que incluye el desempeño de los estudiantes que usan relaciones proporcionales entre todos los datos para obtener la respuesta correcta.
... El primer indicador en este subcomponente contempla la necesidad de presentar de forma clara los conceptos fundamentales de la proporcionalidad, según el nivel educativo; para el caso de secundaria sería necesario distinguir entre razón, tasa, proporción, porcentaje, fracción y número racional, entre otros (Aroza et al., 2016). Este aspecto recibe especial atención en el enfoque aritmético de la proporcionalidad, que tiene un carácter predominante en varias propuestas curriculares e investigaciones (Ben-Chaim et al., 2012;Fernández y Puig, 2002;Hart, 1998;Lamon, 2007;Shield y Dole, 2013). Para Fernández y Puig (2002), el estatuto lógico de la razón es de un nivel más elevado que el de número, fracción, longitud y otros conceptos previos, ya que proviene de una relación de equivalencia. ...
El objetivo de este trabajo es construir una pauta de indicadores de idoneidad didáctica específicos para el tema de la proporcionalidad, que pueda servir de apoyo para el análisis y valoración de lecciones de libros de texto, y como recurso para la reflexión de los profesores sobre procesos de instrucción efectivamente implementados sobre proporcionalidad. Dado que el estudio del razonamiento proporcional se ha constituido en un campo de investigación relevante en educación matemática, es posible identificar criterios de idoneidad específicos para orientar los procesos de enseñanza y aprendizaje, aplicando la metodología de análisis de contenido a una revisión bibliográfica de investigaciones claves en esta área. Las facetas, componentes e indicadores de la noción de idoneidad didáctica orientan la selección y categorización de conocimientos didáctico-matemáticos sobre la proporcionalidad en educación primaria y secundaria derivados de las investigaciones sobre este contenido. Finalmente, se obtiene una Guía de análisis de lecciones de libros de texto de Matemáticas adaptada al tema de proporcionalidad. Dado que el libro de texto es un material curricular que determina en gran medida lo que sucede en el aula y actúa como mediador en el aprendizaje del estudiante, y que el estudio de las razones, proporciones y la proporcionalidad en los currículos de educación primaria y secundaria es un hecho, la guía obtenida puede ser un recurso valioso para el docente. Es necesario el diseño e implementación de acciones formativas con profesores para su conocimiento y uso competente, entendiendo que no supone una propuesta cerrada.
... The importance of the study of ratio, proportion and proportionality in the Primary and Secondary Education curriculum is supported by the decisive role that diverse researchers in mathematics education give to proportional reasoning for developing students' algebraic thinking (Lesh et al. 1988;Van Dooren et al. 2010). However, and despite the importance of this content, there is much evidence that both prospective and in-service teachers have difficulties in understanding and teaching some of the proportional reasoning components (Ben-Chaim et al. 2012;Berk et al. 2009;Buforn et al. 2018;Livy and Vale 2011;Riley 2010;Rivas et al. 2012), as well as to interpret the responses of primary education students when they solve proportionality tasks. In particular, prospective teachers struggle to interpret the responses of primary education students when they solve proportionality tasks, and to consider the way in which students seem to understand proportionality in order to take decisions (Buforn et al. 2020;Fernández et al. 2013;Son 2013). ...
In order to foster the learning of mathematics, the teacher must be able to analyse and assess the students’ mathematical activity. The explicit recognition of objects and processes involved in mathematical practices is a competence that the teacher should develop. This cognitive analysis competence allows the teacher to understand the processes of mathematical learning, to foresee conflicts of meanings and to establish different possibilities for institutionalising the mathematical knowledge involved.
In this article we present the results of the evaluation phase of a training intervention with eighty-eight prospective primary school teachers, which aims to promote and assess the competence for the cognitive analysis of students’ solutions to proportionality tasks. To this end, we proposed the prospective teachers to interpret different students’ solution strategies for a problem, recognise the mathematical elements (languages, concepts, propositions, procedures and arguments) put at stake in each strategy, and to analyse the algebraic character of the mathematical practices involved in them. The results reveal the prospective teachers’ limitations for the analysis and assessment of non-usual resolution strategies, the identification of key mathematical objects and the discrimination of arithmetic and algebraic activity in the students’ solutions. The improvement of the results requires the following actions: to allow prospective teachers to become acquainted with different forms of reasoning that can be applied in proportionality situations, delve more deeply into the algebraic character of mathematical activity, and extend the number and variety of situation problems that prospective teachers can analyse and discuss.
... Although "middle-grade teachers should have a deep understanding of the conceptual components of proportional reasoning and their centrality to all mathematical thinking" (Sowder, Armstrong, Lamon, Simon, Sowder &Thompson, 1998, p. 144), several researches indicate that both prospective and in-service teachers have difficulties to teach concepts related to proportionality (Ben-Chaim, Keret & Ilany, 2012;Berk, Taber, Gorowara & Poetzl, 2009;Hilton & Hilton, 2019;Rivas, Godino & Castro, 2012). The lack of understanding of developing proportional reasoning motivates that teachers often focus on the algorithmic aspect resorting to procedural arguments to justify their strategies in solving proportionality problems (Lamon, 2007;Riley, 2010). ...
Solving problems by different methods, identifying the knowledge put at stake in each case, and stating variants of the problems are fundamental aspects of the competence of analysis of mathematical knowledge for teaching. This paper reports on the design, implementation, and results of a formative intervention with primary education prospective teachers to promote developing this competence using tasks that involve proportional and algebraic reasoning. The experience has been carried out with a sample of 88 students (two class-groups), applying a didactic model that includes work in teams, institutionalization, and assessment of the individual learning achieved. Seventy three percent of students were successful in solving problems; however, only 27% of students managed to solve the four problems proposed by at least two different procedures. More than half of the students adequately identified the knowledge involved in each problem and the algebraization level was correctly assigned in more than half of the proposed solutions. Elaborating meaningful variants to the problems was only achieved in a suitable way by less than 20% of the students. It is concluded that developing the epistemic analysis competence of tasks bringing into play proportional and algebraic reasoning requires a greater attention in the teachers’ formative programs.
... The cross multiplication strategy would be a more suitable alternative to evaluate non-integer ratios (4 × 9 = 6 × 6). Thus, to succeed in both non-integer and integer proportional problems, students need to know about these diverse strategies (e.g., Ben-Chaim, Keret, & Ilany, 2012). ...
We investigated not only the effects of schema-based instruction (SBI) on the mathematical outcomes of seventh-grade students with mathematical learning disabilities (MLD), but also extended prior work to analyze students’ written explanations on open-ended items involving ratio and proportion situations—ratio, proportion, and percent of change problems— to understand the ability to reason about proportions and identify misconceptions. The sample of 338 students with MLD [scored below the 25th percentile on a proportional problem solving (PPS) pretest] was taken from Jitendra, Harwell, Im, et al. (2019), which randomly assigned classrooms to either the SBI or control condition. Students with MLD in SBI classrooms outperformed their counterparts in control classrooms on proportional problem solving and general mathematics problem solving. Similar results, favoring the SBI condition, were found on the open-ended items; however, overall mean scores across pretest, posttest, and delayed posttest were low. Findings provide evidence for the limited understanding of fractional representations of ratios and highlight students’ persistent use of numerical and additive reasoning in explaining their low performance on the open-ended items.
... Measuring Pre-service Teachers' Knowledge for Teaching Fractions Fractions are widely used in mathematics education and have great importance in other disciplines as well (Ben-Chaim, Keret, & Ilany, 2012). For instance, fractions form the basis of introductory mathematics and other mathematical topics such as algebra and probability (Clarke & Roche, 2009). ...
Fractions are a crucial mathematical topic, which is known as being challenging for both students, pre-and in-service teachers. The current study attempted to explore the extent of pre-service middle level mathematics teachers' knowledge for teaching fractions. Fifty-one senior pre-service teachers took part in the study. The Knowledge for Teaching Fractions Test, which includes almost all characteristics of fractions, was employed to measure participants' knowledge for teaching fractions. The Pre-service Teachers' Mathematics Knowledge for Teaching Framework was used to conceptualize participants' responses to the tasks in the Knowledge for Teaching Fractions Test. The findings showed that participants' mathematical knowledge for teaching fractions was not satisfactory. Meanwhile, while they had sound common content knowledge for teaching fractions, their specialized content knowledge for teaching this topic was very poor. The implications for teaching fractions are discussed in the context of pre-service middle level teacher preparation.
... With regard to the study of proportionality in the field of teacher education, there is a growing development of research focused on the study of mathematical knowledge per se and the pedagogical knowledge required to teach proportionality (Izsák & Jacobson, 2013;Sowder et al, 1998). As various works of research show, both teachers in initial and in-service training have difficulties in teaching concepts related to proportionality (Ben-Chaim, Keret & Ilany, 2012, Berk, Taber, Gorowara & Poetzl, 2009, Buforn, Llinares & Fernández, 2018Rivas, Godino & Castro, 2012). The difficulties that teachers have in the primary and secondary stages with the concepts of ratio and proportion, motivate the use of algorithmic procedures such as the rule of three to teach how to solve problems that require proportional reasoning. ...
... Dos estudiantes proponen sendas soluciones recurriendo a la estrategia "valor faltante" (BEN-CHAIM; KERET;ILANY, 2012, p. 138). Una variante en forma de diagrama de esta estrategia de solución es la conocida regla de tres. ...
Resumen En este trabajo se describe el diseño e implementación de una acción formativa con estudiantes de un máster de profesorado de Educación Secundaria en España sobre el tema de proporcionalidad. El objetivo principal de la experiencia es explorar sus conocimientos iniciales y evaluar el grado de desarrollo de aspectos relevantes de la faceta epistémica del conocimiento didáctico-matemático de dicho contenido, en particular, el reconocimiento de niveles de algebrización involucrados en distintas soluciones a problemas de proporcionalidad. En la experiencia han participado 33 estudiantes con titulaciones de grado diversas y se ha realizado en el marco de la asignatura de Iniciación a la Innovación Docente e Investigación en Educación Matemática. Entre los resultados obtenidos destacamos las limitaciones en la competencia de análisis epistémico lograda, en particular, la identificación de proposiciones y procedimientos y sus correspondientes argumentaciones. En algunos casos, los estudiantes han mostrado carencias en el conocimiento común del contenido, el cual sirve de base para el conocimiento didáctico-matemático de este. Se concluye que la mejora de los resultados requiere, entre otras acciones, incrementar el tiempo asignado a la intervención formativa, lo cual permitirá incrementar el número y variedad de situaciones-problema planteadas, su solución y discusión.
... Menurut Lesh (1988) penalaran proporsional adalah bentuk penalaran matematika yang melibatkan kovariasional dan beberapa perbandingan, serta kemampuan mental untuk menyimpan dan memproses beberapa bagian dari informasi. Walle (2013, p.357;Ben-Chaim, Keret, & Ilany;2012:32) menambahkan bahwa penalaran proporsional merupakan "a way of reasoning about multipilcative situation". Dengan kata lain, penalaran proporsional merupakan cara seseorang dalam bernalar pada situasi multiplikatif. ...
Penelitian ini merupakan penelitian deskriptif dengan pendekatan kualitatif. Tujuan penelitian ini adalah mendeskripsikan profil penalaran proporsional siswa SMP dalam memecahkan masalah matematika berdasarkan gaya kognitif sistematis dan intuitif. Penelitian ini dilakukan di kelas IX SMP Al-Muslim Sidoarjo pada tahun ajaran 2016/2017. Subjek penelitian ini adalah seorang siswa bergaya kognitif sistematis dan seorang siswa bergaya kognitif sistematis. Alat pengumpul data berupa Tes Gaya Kognitif (TGK), Tugas Pemecahan Masalah (TPM), Pedoman wawancara, dan alat rekam audio visual. Data penelitian diperoleh dari pemberian tugas pemecahan masalah dan wawancara sebanyak dua kali. Wawancara dilakukan untuk mengungkapkan profil penalaran proporsional dalam memecahkan masalah matematika terkait materi perbandingan. Keabsahan data diuji dengan triangulasi waktu. Hasil penelitian menunjukkan kedua subjek gagal membedakan masalah proporsional dan bukan proporsional pada tahap memahami masalah. Pada tahap menyusun rencana, Siswa bergaya kognitif sistematis mengelompokkan bagian-bagian yang sebanding untuk membuat persamaan. Sedangkan siswa bergaya kognitif intuitif membandingkan luas lahan pertama dan kedua, kemudian menyederhanakan perbandingannya. Pada tahap melaksanakan rencana, siswa bergaya kognitif sistematis menggunakan strategi cross product algorithm untuk menemukan solusi. Sedangkan siswa bergaya kognitif intuitif menggunakan strategi build-up method dan factor of change untuk menemukan solusi. Pada tahap memeriksa kembali, Siswa bergaya konitif sistematis dan intuitif mengecek solusi yang mereka peroleh dengan mensubtitusi masing-masing solusi ke persamaan, kemudian melihat nilai perbandingan yang dihasilkan. Jawaban benar jika nilai perbandingan dalam persaamaan tersebut sama. This research uses descriptive qualitative approach. The purpose of this study was to describe proportional reasoning profile of junior high school student in mathematical problem solving based on systematic and intuitive cognitive style. This study was do ini class IX SMP Al-Muslim Sidoarjo in 2016/2017. One subject is systematic cognitive style and the other is intuitive cognitive style. The colect data instruments are TGK, TPM, interview guidence, and video recorder. Data were obtain from giving problem solving task and gave interview twice during the study. Interviews were do for exploration proportional reasoning profile of junior high school student in mathematical problem solving about proportional problems. The validity of the data is tested using time triangulation. The result of this research showed both of subjects failed to different the proportional problem and non-proportional problems on understanding step. On planning step, systematic student collected the same parts to make a equation. Intuitive student compared large of first field and second, and simplified it. On doing step, systematic student use cross product algorithm strategy to find solution. Intuitive student used build-up method and factor of change strategy to find solution. On overview step, systematic and intuitive student checking with substitution the solution in equation and look at the comparing values. If the comparing values are same, it means the subject got correct answer.
The aim of this paper is to describe and analyze how a group of prospective teachers create problems to develop proportional reasoning either freely or from a given situation across different contexts, and the difficulties they encounter. Additionally, it identifies their beliefs about what constitutes a good problem and assesses whether these beliefs are reflected in their problem creation. This is a descriptive-qualitative study that utilizes theoretical and methodological tools from the Onto-semiotic Approach in the content analysis of participants' responses. The results indicate that the prospective teachers' beliefs about what makes a good problem do not always manifest in their practice. The prospective teachers faced challenges in inventing problems that meet the established didactic-mathematical purpose, related to insufficient didactic-mathematical knowledge of proportional reasoning, achieving better outcomes in the arithmetic context and in free creation.
In this pilot study, we extend existing research in the field of mathematics-focused maker-based STEM education. Specifically, we report on the ways in which seventh-grade students represented their knowledge of proportional reasoning as they created digital 3-dimensional models of their dream homes in a school makerspace STEM integrated activity. Using a directed content analysis, we examined written, verbal, and visual data collected from the students across the project. Results indicated that students represented their understanding of proportional reasoning through open-ended written responses more accurately than through verbal descriptions or digital artifacts. In addition, the geometric and numeric dimensions of proportional reasoning, and their respective components were represented more often by students than any other dimension of proportional reasoning across the data. This suggests a maker-based instructional approach to teaching proportional reasoning in the middle grades may help students gain a solid foundation in those mathematical components and twenty-first century skills such as communication and collaboration. Implications and recommendations for practice and research are discussed.
The aim of this research is to examine the problems related to the concept of ratio-proportion in Higher Education Institutions Exams (HIE) in the last 57 years from 1966 to 2022 based on Lithner's (2008) mathematical reasoning framework. The design of this study is considered in the context of the analytical research model. Moreover, data were collected through document review in this study. The data of this study were analyzed by descriptive analysis approach based on Lithner's (2008) mathematical reasoning framework. From 1966 to 2022, 164 mathematical problems regarding the concept of ratio-proportion were identified in the HIE. Research findings showed that 84% of the ratio-proportional problems in the HIE in the last 57 years can be solved by making imitative reasoning (IR), while only 16% can be solved by making creative reasoning (CR). In terms of imitative and creative reasoning components, most of problems [70%] can be solved by making algorithmic reasoning (ALGR), while very few of these problems [3%] can be solved by making global creative reasoning (GCR). The results indicate that 14% of these problems are solved by making memorized reasoning (MR) and 70% by making ALGR in the context of IR. Additionally, it has been revealed that 13% of these problems are solved by making local creative reasoning (LCR) and 3% of these problems are solved by making global creative reasoning (GCR) in the context of CR. The results of this research indicated that students need to make creative reasoning instead of imitative reasoning in the HIE that has taken place in recent years. Since this research examines in depth the problems regarding the concept of ratio-proportion in the HIE from 1966 to 2022 in terms of mathematical reasoning, it is thought that the research findings provide useful information to mathematics education researchers and mathematics teachers in our country.
(Objetivo)
El objetivo de este artículo es describir y analizar una intervención formativa con futuro personal docente de primaria, dirigida a desarrollar la competencia para crear problemas de proporcionalidad mediante la variación de un problema inicial y la elaboración a partir de un requerimiento didáctico-matemático.
(Metodología)
Se trata de un estudio cualitativo e interpretativo que adopta una metodología propia de las investigaciones de diseño o ingeniería didáctica. Tanto en el diseño de la intervención, como en el análisis de contenido de las respuestas de los sujetos participantes se emplean herramientas teóricas y metodológicas del enfoque ontosemiótico. Se trabaja con un grupo de 127 estudiantes para docentes de Educación Primaria de la Universidad de Granada, España; organizados en 33 equipos para responder a dos consignas sobre creación de problemas.
(Resultados)
Los resultados muestran que los sujetos participantes crean con mayor frecuencia problemas pertinentes mediante la variación de un problema dado, pero que no logran crear problemas que permitan, de manera específica, distinguir situaciones proporcionales de aditivas a partir de un requerimiento didáctico-matemático.
(Conclusiones)
Se concluye que los futuros maestros y maestras manifiestan un conocimiento didáctico y matemático insuficiente para crear problemas de proporcionalidad exitosamente. Por esto es necesario que los programas de formación refuercen las estrategias para desarrollar esta tarea, incorporándola como recurso didáctico en el proceso de enseñanza, y mejorando la competencia para el análisis de la actividad matemática del futuro profesorado.
(Objetivo)
El presente trabajo tiene como objetivo describir los resultados de la evaluación de los conocimientos y competencias de futuros profesores de Educación Primaria para interpretar las respuestas de estudiantes a tareas de comparación de probabilidades, identificar estrategias incorrectas y reconocer razonamiento proporcional en la actividad matemática. Asimismo, se analizan las distintas formas de actuación que proponen los futuros docentes con objeto de que los alumnos superen las dificultades que los llevaron a dar una solución inadecuada.
(Metodología)
Para tal fin, se ha realizado una investigación de carácter descriptivo y cualitativo, para la cual se ha contado con la colaboración de 116 futuros profesores de Educación Primaria de la Universidad de Almería (España). La intervención se llevó a cabo una vez finalizado el proceso de formación en torno a los contenidos matemáticos del Bloque de Estadística y Probabilidad.
(Resultados)
Entre los resultados obtenidos destacamos un conocimiento didáctico-matemático del razonamiento proporcional y probabilístico que impide a los futuros profesores interpretar y tomar decisiones en relación con las respuestas de los alumnos.
(Conclusiones)
Estos resultados ponen de manifiesto la necesidad de implementar intervenciones formativas que permitan solventar adecuadamente estas situaciones habituales en los centros educativos.
Los estadísticos de orden tienen gran importancia en el análisis exploratorio de datos e inferencia estadística no paramétrica y se estudian en diversos niveles educativos. La investigación didáctica describe errores en su comprensión. En este trabajo analizamos la relación de los estadísticos de orden con el razonamiento proporcional, apoyándonos en el enfoque ontosemiótico, y la investigación sobre estadístico de orden y razonamiento proporcional. Se analizan ejemplos que muestran la relación de los errores asociados a los estadísticos de orden con el razonamiento proporcional. Se finaliza con algunas implicaciones para la mejora de la enseñanza de estos estadísticos.
Penelitian ini bertujuan untuk mendeskripsikan literasi matematika siswa kelas IX SMP di Sidoarjo dalam menyelesaikan soal Higher Order Thinking Skills (HOTS) pada topik proporsi. Penelitian ini termasuk penelitian deskriptif kualitatif dengan menggunakan metode tes tulis terkait soal HOTS proporsi dan wawancara. Data yang diperoleh dianalisis menggunakan indikator literasi matematika sesuai pada Kerangka Kerja PISA 2021 pada tahap merumuskan (formulate), menerapkan (employ), serta menafsirkan dan mengevaluasi (interpret and evaluate). Subjek penelitian dipilih menggunakan teknik purposive sampling. Dari 12 siswa dengan kemampuan matematis tinggi, tiga siswa dengan literasi matematika yang berbeda dipilih sebagai subjek pada penelitian ini. Hasil penelitian menunjukkan bahwa dalam menyelesaikan soal HOTS proporsi, pada tahap merumuskan (formulate), siswa kurang mampu merepresentasikan situasi matematis menggunakan model matematika yang sesuai dengan topik proporsi. Pada tahap menerapkan (employ), siswa mampu menggunakan konsep dan prosedur matematis untuk menyelesaikan soal HOTS proporsi. Sedangkan, pada tahap menafsirkan dan mengevaluasi (interpret and evaluate), siswa kurang mampu menafsirkan hasil matematis kembali ke konteks dunia nyata. Sehingga, diperlukan pembelajaran yang bukan hanya melatih siswa untuk menerapkan konsep dan prosedur matematis untuk menyelesaikan soal, namun juga dapat melatih siswa untuk merepresentasikan situasi matematis dalam konteks dunia nyata menjadi model matematika dan menafsirkan solusi matematis yang telah diperoleh kembali ke konteks dunia nyata. Kata Kunci : Literasi matematika, Soal HOTS, Proporsi
The purpose of this study was to investigate prospective middle school mathematics teachers’ proportional reasoning before and after receiving a practice-based instruction based on proportional reasoning. The Proportional Reasoning Test, semi-structured interviews and observations of student teachings were used to collect data about the participants’ proportional reasoning. The results indicated that prospective teachers improved their proportional reasoning by completing a practice-based instruction. Before the instruction, they generally applied algebraic procedures without associating meaning and used a limited number of strategies to solve problems. Furthermore, they had difficulties in distinguishing proportional from nonproportional situations. However, by the end of the instruction, while they mostly preferred to use informal strategies, they relied less on formal strategies. Additionally, they utilized a broader range of strategies to solve problems and made sense of these strategies. Further, they could determine whether the quantities in a situation were related additively, multiplicatively, or in some other way.
A pesar de los enormes esfuerzos de investigación que se vienen realizando, el problema de cómo enseñar las matemáticas y las ciencias sigue abierto. Decidir entre los modelos didácticos centrados en el profesor (enseñanza transmisiva) o centrados en el estudiante (aprendizaje indagativo) plantea un dilema para la práctica educativa. En este trabajo abordamos este problema y proponemos una posible solución aplicando los supuestos y herramientas teóricas del Enfoque Ontosemiótico. Se argumenta que la optimización del aprendizaje y el logro de una acción didáctica idónea requiere entretejer de manera dialéctica y compleja los momentos de transmisión del conocimiento por el profesor con los momentos de indagación del estudiante. La implementación de trayectorias didácticas eficientes por parte del docente, implica la articulación de diversos tipos de configuraciones didácticas gestionadas mediante criterios de idoneidad, los cuales deben tener en cuenta las dimensiones epistémica, cognitiva, afectiva, interaccional y mediacional.Palabras clave: Modelos didácticos, objetivismo, enfoque ontosemiótico, idoneidad didáctica.Como ensinar a matemática e as ciências experimentais?Resolvendo o dilema entre transmissão e indagaçãoResumoApesar dos enormes esforços de investigação que se vêm realizando, o problema de como ensinar matemática e ciências permanece em aberto. Decidir ente um modelo didático centrado no professor (ensino transmissivo) ou um modelo didático centrado no estudante (aprendizagem indagativa), representa um dilema para a prática educacional. Neste trabalho, abordamos este problema e propomos uma solução aplicando os pressupostos teóricos e ferramentas do Enfoque Ontossemiótico. Argumenta-se que a otimização da aprendizagem e a realização de uma ação didática idónea requer um cruzamento dialético e complexo dos momentos de transmissão do conhecimento pelo professor com os momentos de indagação do estudante. A implementação de trajetórias didáticas eficientes implica por parte do professor a articulação de diversos tipos de configurações didáticas orientadas por critérios de idoneidade didática, os quais devem ter em consideração as dimensões epistêmica, cognitiva, afetiva e interacional.Palavras chave: Modelos didáticos, objetivismo, enfoque ontossemiótico, idoneidade didática.How to teach mathematics and experimental sciences?Solving the inquiring versus transmission dilemmaAbstractDespite the huge research efforts that have been made, the problem of how to teach mathematics and sciences remains open. Deciding between teacher-focused teaching models (transmissive teaching) or student-focused (inquiring learning) poses a dilemma for educational practice. In this paper we address this problem and propose a solution applying the Onto-semiotic Approach assumptions and theoretical tools. We argue that the learning optimization and achievement of an appropriate didactic intervention require interweaving in a dialectical and complex way, the teacher’s moments of knowledge transmission with the student’s inquiry moments. The implementation of efficient didactic trajectories implies the articulation of diverse types of didactic configurations managed through didactical suitability criteria on the teacher´s part. These should take into account the epistemic, cognitive, affective, interactional, mediational and ecological dimensions involved in instructional processes.Keywords: didactical models, constructivism, objectivism, onto-semiotic approach, didactical suitability
A pesar de los enormes esfuerzos de investigación que se vienen realizando, el problema de cómo enseñar las matemáticas y las ciencias sigue abierto. Decidir entre los modelos didácticos centrados en el profesor (enseñanza transmisiva) o centrados en el estudiante (aprendizaje indagativo) plantea un dilema para la práctica educativa. En este trabajo abordamos este problema y proponemos una posible solución aplicando los supuestos y herramientas teóricas del Enfoque Ontosemiótico. Se argumenta que la optimización del aprendizaje y el logro de una acción didáctica idónea requiere entretejer de manera dialéctica y compleja los momentos de transmisión del conocimiento por el profesor con los momentos de indagación del estudiante. La implementación de trayectorias didácticas eficientes por parte del docente, implica la articulación de diversos tipos de configuraciones didácticas gestionadas mediante criterios de idoneidad, los cuales deben tener en cuenta las dimensiones epistémica, cognitiva, afectiva, interaccional y mediacional.Palabras clave: Modelos didácticos, objetivismo, enfoque ontosemiótico, idoneidad didáctica.Como ensinar a matemática e as ciências experimentais?Resolvendo o dilema entre transmissão e indagaçãoResumoApesar dos enormes esforços de investigação que se vêm realizando, o problema de como ensinar matemática e ciências permanece em aberto. Decidir ente um modelo didático centrado no professor (ensino transmissivo) ou um modelo didático centrado no estudante (aprendizagem indagativa), representa um dilema para a prática educacional. Neste trabalho, abordamos este problema e propomos uma solução aplicando os pressupostos teóricos e ferramentas do Enfoque Ontossemiótico. Argumenta-se que a otimização da aprendizagem e a realização de uma ação didática idónea requer um cruzamento dialético e complexo dos momentos de transmissão do conhecimento pelo professor com os momentos de indagação do estudante. A implementação de trajetórias didáticas eficientes implica por parte do professor a articulação de diversos tipos de configurações didáticas orientadas por critérios de idoneidade didática, os quais devem ter em consideração as dimensões epistêmica, cognitiva, afetiva e interacional.Palavras chave: Modelos didáticos, objetivismo, enfoque ontossemiótico, idoneidade didática.How to teach mathematics and experimental sciences?Solving the inquiring versus transmission dilemmaAbstractDespite the huge research efforts that have been made, the problem of how to teach mathematics and sciences remains open. Deciding between teacher-focused teaching models (transmissive teaching) or student-focused (inquiring learning) poses a dilemma for educational practice. In this paper we address this problem and propose a solution applying the Onto-semiotic Approach assumptions and theoretical tools. We argue that the learning optimization and achievement of an appropriate didactic intervention require interweaving in a dialectical and complex way, the teacher’s moments of knowledge transmission with the student’s inquiry moments. The implementation of efficient didactic trajectories implies the articulation of diverse types of didactic configurations managed through didactical suitability criteria on the teacher´s part. These should take into account the epistemic, cognitive, affective, interactional, mediational and ecological dimensions involved in instructional processes.Keywords: didactical models, constructivism, objectivism, onto-semiotic approach, didactical suitability
The current study investigated the relationship between students’ mathematical thinking style and their modeling processes and routes. Thirty-five eighth-grade students were examined. In the first stage, the students solved questions, and according to their solutions, they were assigned to one of two thinking style groups: visual and analytic. The two groups engaged in three modeling activities. Findings indicated differences in the groups’ modeling processes in performing the three activities. The primary differences in the modeling processes were manifested in simplifying, mathematizing, and eliciting a mathematical model. In addition, the analytic thinking group skipped the real-model phase in the three activities, while the visual group built a real model for each activity.
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.
Aunque suele haber un consenso bastante generalizado en educación matemática a favor de los modelos de instrucción de tipo constructivista la cuestión de su pertinencia no deja de ser controvertida. Entre los modelos extremos centrados, bien en el estudiante o en el profesor, se pueden encontrar otros modelos de tipo mixto en los que ambos agentes del proceso educativo tienen papel protagonista, dependiendo del contenido cuyo aprendizaje
se pretende y de los conocimientos previos de los estudiantes. En este trabajo se describe y fundamenta la implementación de un modelo instruccional de tipo mixto que contempla una primera fase en la que el profesor adquiere el papel protagonista introduciendo el tema, una segunda fase de trabajo colaborativo entre profesor y
alumnos, en la que resuelven conjuntamente una situación-problema, seguida de una tercera fase en la que los alumnos trabajan de manera autónoma. Este modelo ha sido experimentado con alumnos de 5º curso de primaria, siendo su objetivo crearles un primer encuentro con los problemas de proporcionalidad directa. Aunque se trata de un estudio de caso que no permite generalizar los resultados, la evaluación de los aprendizajes logrados permite formular hipótesis sobre la influencia del modelo mixto de instrucción en los aprendizajes de los alumnos, las cuales se pueden contrastar en nuevos ciclos de investigación sobre este tema y en contextos similares.
Aroza, C. J., Godino, J. D., & Beltrán-Pellicer, P. (2016). Iniciación a la innovación e investigación educativa mediante el análisis de la idoneidad didáctica de una experiencia de enseñanza sobre proporcionalidad. AIRES, 6(1).
Resumen La innovación fundamentada de los procesos de enseñanza y aprendizaje de las matemáticas requiere del profesor una actitud y competencia para la reflexión e indagación sistemática sobre la propia práctica. El desarrollo de dicha competencia debe ser un objetivo de la formación inicial de profesores. En este artículo se describe el proceso de indagación y reflexión sobre una experiencia de enseñanza realizada en la fase de prácticas del máster de formación inicial de profesorado de secundaria en la especialidad de Matemáticas. La reflexión se realiza aplicando la noción de idoneidad didáctica a un proceso de enseñanza y aprendizaje implementado sobre la proporcionalidad y porcentajes en primer curso de educación secundaria. La valoración de la idoneidad didáctica, y la consiguiente identificación de propuestas fundamentadas de cambio para el rediseño de la experiencia, requiere recopilar y sintetizar los conocimientos didáctico-matemáticos producidos en la investigación e innovación sobre la enseñanza y aprendizaje de la proporcionalidad. Dichos conocimientos son sintetizados en criterios de idoneidad específicos para el tema abordado. Se concluye que la aplicación de los criterios de idoneidad didáctica ayuda a sistematizar los conocimientos didácticos y su aplicación a la reflexión y mejora progresiva de la práctica de la enseñanza. Palabras Clave Educación matemática, formación de profesores, educación secundaria, idoneidad didáctica, proporcionalidad y porcentajes.
This study aims to explore how Indonesian students make sense direct proportion concepts according to their level of mathematics anxiety. This research tried to uncover the meaning of a phenomenon for students who are involved. In represents the findings, this research emphasized on seeing through the eyes of students being studied. Fifty-six sixth grader students were asked to complete the CAMS instrument. After completing Indonesian version of Children's Anxiety in Math Scale (CAMS), three female students' mathematics anxiety levels were created (high, medium, and low). High anxiety student tends to fail in doing an understanding. Student used additive relationships, response without reasons, or use numbers, operations, or strategies randomly in solving the given problems. Medium anxiety student tends to connect the given information listed in the problem with the concepts that she has already had. Student used qualitative comparison, use rate as a unit, and response without reasons in solving the given problems. Low anxiety student connected the given information listed in the problem with the concepts that she has already had. Student used use ratio as a unit, make qualitative comparison, use cross-product rule, and use symbols, commonly algebraic symbols in representing proportions with full understanding of its relationships
The vast number of online educational videos available at the moment has generated an emerging area of research concerning their level of suitability. This study considers the epistemic quality of educational videos on mathematics, focusing on the specific content of directly proportional distributions. A qualitative study is used, based on the application of theoretical and methodological tools from the onto-semiotic approach to knowledge and mathematics instruction, principally the notion of epistemic suitability and the identification of algebraic levels. The sample consists of the 31 most popular videos in Spanish on YouTube™ on directly proportional distributions. Analysis reveals interesting results on these kinds of resources. In general, it is observed that they are weak in epistemic suitability, which does not seem to affect their level of popularity. Moreover, the existence of videos with inaccurate arguments or incorrect procedures, together with the diversity of algebraic levels used, indicates that teachers should be careful when selecting them and only recommend those that better suit their students’ needs.
En este trabajo se realiza una síntesis de conocimientos didáctico-matemáticos sobre el estudio de la proporcionalidad en educación primaria y secundaria. Utilizamos como marco teórico la Teoría de la Idoneidad Didáctica, desarrollada dentro del Enfoque Ontosemiótico del Conocimiento y la Instrucción Matemáticos. El sistema de categorías de facetas y componentes, así como los criterios o indicadores generales de idoneidad que propone dicha teoría, son aplicados para analizar y clasificar los resultados de investigaciones relevantes sobre la enseñanza y aprendizaje de la proporcionalidad y nociones relacionadas. Se propone finalmente un sistema de criterios de idoneidad para la faceta epistémica (conocimientos institucionales) específicos para el estudio del tema. Estos indicadores se pueden utilizar en la formación de profesores, así como para valorar la idoneidad didáctica de recursos y experiencias de enseñanza. 1. Introducción Si bien no se pueden esperar de las didácticas especiales recetas generales para la enseñanza de los contenidos curriculares que indiquen qué y cómo enseñar en cada circunstancia, es razonable pensar que de los esfuerzos de investigación se deriven resultados que orienten y ayuden a los profesores en las tareas docentes. Este es uno de los supuestos subyacentes de la Teoría de la Idoneidad Didáctica (TID) (Godino, Bencomo, Font y Wilhelmi, 2006; Godino, 2013), mediante la cual se propone sistematizar principios, indicadores, criterios o heurísticas, sobre los cuales existe un consenso en la comunidad educativa de un campo específico, cuya aplicación podría ayudar a alcanzar niveles altos de idoneidad de los procesos instruccionales.
Temperature, velocity, acceleration, and density are but a few examples of scientific concepts that are expressed by a ratio. These topics are difficult for students to understand because they do not understand the mathematical concept of a ratio.
In this article we study the concept of invariance of ratio through an investigation of children's understanding of constancy of taste--that is, the notion that random samples of a given mixture taste the same--using a device that does not resort to conventional symbolism. The paper begins with a definition of constancy of taste and other quantitative analogues. Then it presents a theoretical analysis of how constancy of taste may emerge from the child's additive world and grow into a conception where taste becomes an intensive quantity. The analysis suggests that one's conception of taste constancy is linked in a fundamental way to one's conception of invariance of ratio. Following this analysis, the paper reports a study that demonstrates the absence of taste constancy among sixth-grade children. More specifically, the study shows that sixth-grade children base their judgment of the relative strength of the taste of two samples from the same mixture on at least one of three (extraneous) variables: the relative volumes of the samples to be tasted, whether the mixture is thought of as consisting of a single ingredient or more than one ingredient, and the relative amount of the ingredients stated in the problem.
This paper reports on a project designed to develop inquiry communities between teachers and didacticians, aimed at improving the learning of mathematics in classrooms, and at studying the processes, practices, issues and outcomes in and of the project. Theoretical notions of inquiry and community underpin the project. The focus here is largely methodological, tracing the origination and development of the project and decisions taken through its first phases. The project is seen to be situated within a 'developmental' research paradigm in which research both studies the developmental process and contributes centrally to it. Issues in the interpretation of inquiry practices in mathematics learning and teaching and the building of communities at various levels are seen as important outcomes. The roles and relationships of teachers and didacticians emerge as key concepts in the developmental process.
This article describes and shares an innovative pedagogical practice that holds promise in contributing to the teaching and learning of proportions in middle school. The teaching and learning of mathematics with understanding framework was used as a vehicle to help 21 seventh grade students reason proportionally. The findings of this unit suggest that the classroom culture, which encouraged the students to make connections between their existing and new ideas and reflecting and communicating their thinking, contributed to their emerging understanding of proportions. The use of an authentic and non-routine task involving liquid measurements also heightened their interest, curiosity and enthusiasm, thereby contributing to their excitement about the mathematics they were learning.
This study focused on 5th‐grade teachers' and students' beliefs about mathematical problem solving; attributions for the causes of performance in problem solving; and beliefs about the teaching and learning of problem solving in mathematics. Ten 5th‐grade teachers from four different schools in a large rural school district volunteered to participate in the study. Each teacher identified two students from her classroom to participate in the study, one student perceived by the teacher as successful and one perceived as unsuccessful in mathematical problem solving. Parallel interviews with teacher and students were conducted. During the interviews, a set of nine word problems was presented, and teachers predicted the likelihood that the participating students would get correct answers on each of the problems. Students were asked to solve each of the problems orally. Verbatim transcripts of the interviews were analyzed and a correlation of teachers' predictions of students' performance and the students' actual performance was considered.
Four general conclusions resulted from the research. First, 5th‐grade teachers believe that problem solving in mathematics in primarily an application of computational skills. Students' beliefs about mathematical problem solving are, for the most part, consistent with the beliefs held by the teachers. Second, 5th‐grade teachers in the study basically attributed success and failure in problem solving to differences in students' ability, while students attributed success and failure to a combination of ability and effort. Third, the activities in problem solving in the 5th‐grade classrooms, if anything, enhance computational ability. The teachers' focus was primarily on right answers, and the use of calculators for problem solving was strongly discouraged. Finally, teachers tend to overestimate the students' ability to do problems involving computation and underestimate students' ability to do reasoning problems.
Although children partition by repeatedly halving easily and spontaneously as early as the age of 4, multiplicative thinking is difficult and develops over a long period in school. Given the apparently multiplicative character of repeated halving and doubling, it is natural to ask what role they might play in the development of multiplicative thinking. We investigated this question by examining children's solutions to folding tasks, which involved predicting the number of equal parts created by a succession of given folds and determining a sequence of folds to create a given number of equal parts. Analyzing a combination of cross-sectional data and case studies from standardized clinical interviews, we found that children were most successful at coordinating folding sequences with multiplicative thinking when they used a conceptualization of doubling based upon recursion. This conceptualization tended to generate more sophisticated solutions.