This document provides an overview of the field of mathematics education from multiple perspectives. It discusses the complex nature of mathematics as both a product of abstraction and a creative process. Mathematics education aims to develop students' mathematical dispositions and ability to assess real-world mathematizations. Research in mathematics education reflects the diversity of perspectives and uses qualitative methods to study learning from individual and social-cultural views. The goals of research are to advance knowledge and practice while addressing issues of complexity, perspective, and standards of inquiry in this dynamic field.
The collective construction of Knowledge - reportalicevborg
This study investigated how 7th grade students' mathematical understanding of probability developed through their participation in different social configurations within a computer-mediated classroom. The researcher examined how whole-class discussions and local investigations using a computer simulation collectively contributed to the students' probabilistic reasoning. Through two case studies, the researcher found that students' mathematical practices developed both for individual cognitive purposes and social communicative purposes of resolving disputes. The analysis showed how different social configurations built upon each other to shape individual learning and foster a shared understanding of probability within the classroom community.
Dr. William Allan Kritsonis, Educational PhilosophyWilliam Kritsonis
The document discusses implementing the six realms of meaning from Kritsonis' Ways of Knowing Through the Realms of Meaning framework in professional learning communities. It examines how each realm - symbolics, empirics, esthetics, synnoetics, ethics, and synoptics - can be applied in PLCs. For example, symbolics focuses on communication, empirics on collecting and analyzing student data, and ethics emphasizes making moral decisions. Overall, the document argues that applying the realms of meaning can help PLCs engage in more holistic and effective dialogue to improve student achievement.
Preservice teachers examine gender equity in teaching mathematicsVirginia Acey
This document discusses gender inequity in mathematics teaching and describes a project used with preservice teachers to help them recognize inequitable practices. It notes that while math scores have risen for both males and females, a persistent gender gap remains. Some stereotypes view females as less capable in math, but research shows all students can succeed with high-quality instruction. The Equity Teaching Analysis Project analyzes actual teaching to help preservice teachers see the need to make instruction more equitable by fostering equal participation from both male and female students.
1) The document discusses the relationship between scaffolding and Vygotsky's concept of the zone of proximal development (ZPD). While scaffolding is commonly seen as operationalizing the ZPD, different scholars have interpreted their relationship in various ways.
2) The ZPD refers to the difference between what a learner can do independently and with guidance. It emphasizes social interaction and collaboration between learners and teachers. Some see scaffolding as only partially reflecting the richness of the ZPD.
3) A narrow view of scaffolding as one-directional instruction could suppress student initiative, going against Piagetian and Vygotskian views of the active learner. A deeper understanding of scaffolding's theoretical basis in
This article provides an overview of key concepts from sociocultural theory and how they can be applied to second language learning research. It discusses four main concepts:
1. Mediation - Learning is viewed as a socially mediated process where language and other tools mediate cognition. Interaction with more capable individuals also mediates learning.
2. Zone of proximal development - The ZPD represents what students can achieve with assistance that exceeds their independent abilities. Social interaction within the ZPD activates learning and development.
3. Scaffolding - Support provided within the ZPD, such as simplifying tasks or demonstrating solutions, allows students to accomplish tasks they cannot yet do independently. Peer scaffolding can also facilitate language development.
INTRODUCING GEOGRAPHY AND READING MAP SKILLS TO KINDERGARTEN CHILDREN BY USIN...ijejournal
The construction of space in children develops gradually, as they grow up and is related both to their
environment and to their spatial and/or geographical experiences. According to previous studies, spatial
thinking is malleable, and can be developed with the use of appropriate teaching interventions and
educational material. Geospatial thinking and reading map skills required to decode map symbols are a
relatively new and very interesting topic in kindergarten’s education. Significance of this study is the
creation of two large-scale giant maps, laminated, and appropriate to accompany teaching material that
can be used in a teaching intervention based on the Greek kindergarten curriculum. The first map (scale of
1:1000) is a map of the city of Mytilene, and the second one (scale of 1:20000) is of the island of Lesvos;
both have dimensions 3X4 meters. The purpose of this study is to present the creation of spatial teaching
material, so that map skills in Kindergarten education can be cultivated and developed in order to provide
kindergarten teachers with large scale-giant maps and encourage them to introduce them to their teaching,
as large scale-giant maps are really powerful educational tools for the development of their pupils’ spatial
skills.
This chapter introduces the concept of mediating artefacts, which are representations that mediate the learning design process. Mediating artefacts can take various forms like narratives, visual models, or vocabularies. They are derived from learning activities through abstraction and highlight different aspects of the learning activity. Mediating artefacts can then be aggregated into meta-mediating artefacts like repositories, scaffolds, or mixed formats to further support the design process. Understanding learning activities through mediating artefacts allows practitioners to make informed choices in designing new learning activities.
This document summarizes a study that examines changes in the relationship between student self-concept and mathematics achievement in Hong Kong during its political transition from British to Chinese rule using TIMSS data from 1995, 1999, and 2003. It identifies two indicators of self-concept - how well students feel they do in math (internal comparison) and how difficult math is for them compared to others (external social comparison). It also discusses how parental education, a factor influenced by population migration during the transition, relates to both self-concept and achievement and may have shown a non-monotonic trend over the time period examined. The study aims to investigate how parental education changes relate to self-concept and achievement development and examine whether the reciprocal relationship model between
The collective construction of Knowledge - reportalicevborg
This study investigated how 7th grade students' mathematical understanding of probability developed through their participation in different social configurations within a computer-mediated classroom. The researcher examined how whole-class discussions and local investigations using a computer simulation collectively contributed to the students' probabilistic reasoning. Through two case studies, the researcher found that students' mathematical practices developed both for individual cognitive purposes and social communicative purposes of resolving disputes. The analysis showed how different social configurations built upon each other to shape individual learning and foster a shared understanding of probability within the classroom community.
Dr. William Allan Kritsonis, Educational PhilosophyWilliam Kritsonis
The document discusses implementing the six realms of meaning from Kritsonis' Ways of Knowing Through the Realms of Meaning framework in professional learning communities. It examines how each realm - symbolics, empirics, esthetics, synnoetics, ethics, and synoptics - can be applied in PLCs. For example, symbolics focuses on communication, empirics on collecting and analyzing student data, and ethics emphasizes making moral decisions. Overall, the document argues that applying the realms of meaning can help PLCs engage in more holistic and effective dialogue to improve student achievement.
Preservice teachers examine gender equity in teaching mathematicsVirginia Acey
This document discusses gender inequity in mathematics teaching and describes a project used with preservice teachers to help them recognize inequitable practices. It notes that while math scores have risen for both males and females, a persistent gender gap remains. Some stereotypes view females as less capable in math, but research shows all students can succeed with high-quality instruction. The Equity Teaching Analysis Project analyzes actual teaching to help preservice teachers see the need to make instruction more equitable by fostering equal participation from both male and female students.
1) The document discusses the relationship between scaffolding and Vygotsky's concept of the zone of proximal development (ZPD). While scaffolding is commonly seen as operationalizing the ZPD, different scholars have interpreted their relationship in various ways.
2) The ZPD refers to the difference between what a learner can do independently and with guidance. It emphasizes social interaction and collaboration between learners and teachers. Some see scaffolding as only partially reflecting the richness of the ZPD.
3) A narrow view of scaffolding as one-directional instruction could suppress student initiative, going against Piagetian and Vygotskian views of the active learner. A deeper understanding of scaffolding's theoretical basis in
This article provides an overview of key concepts from sociocultural theory and how they can be applied to second language learning research. It discusses four main concepts:
1. Mediation - Learning is viewed as a socially mediated process where language and other tools mediate cognition. Interaction with more capable individuals also mediates learning.
2. Zone of proximal development - The ZPD represents what students can achieve with assistance that exceeds their independent abilities. Social interaction within the ZPD activates learning and development.
3. Scaffolding - Support provided within the ZPD, such as simplifying tasks or demonstrating solutions, allows students to accomplish tasks they cannot yet do independently. Peer scaffolding can also facilitate language development.
INTRODUCING GEOGRAPHY AND READING MAP SKILLS TO KINDERGARTEN CHILDREN BY USIN...ijejournal
The construction of space in children develops gradually, as they grow up and is related both to their
environment and to their spatial and/or geographical experiences. According to previous studies, spatial
thinking is malleable, and can be developed with the use of appropriate teaching interventions and
educational material. Geospatial thinking and reading map skills required to decode map symbols are a
relatively new and very interesting topic in kindergarten’s education. Significance of this study is the
creation of two large-scale giant maps, laminated, and appropriate to accompany teaching material that
can be used in a teaching intervention based on the Greek kindergarten curriculum. The first map (scale of
1:1000) is a map of the city of Mytilene, and the second one (scale of 1:20000) is of the island of Lesvos;
both have dimensions 3X4 meters. The purpose of this study is to present the creation of spatial teaching
material, so that map skills in Kindergarten education can be cultivated and developed in order to provide
kindergarten teachers with large scale-giant maps and encourage them to introduce them to their teaching,
as large scale-giant maps are really powerful educational tools for the development of their pupils’ spatial
skills.
This chapter introduces the concept of mediating artefacts, which are representations that mediate the learning design process. Mediating artefacts can take various forms like narratives, visual models, or vocabularies. They are derived from learning activities through abstraction and highlight different aspects of the learning activity. Mediating artefacts can then be aggregated into meta-mediating artefacts like repositories, scaffolds, or mixed formats to further support the design process. Understanding learning activities through mediating artefacts allows practitioners to make informed choices in designing new learning activities.
This document summarizes a study that examines changes in the relationship between student self-concept and mathematics achievement in Hong Kong during its political transition from British to Chinese rule using TIMSS data from 1995, 1999, and 2003. It identifies two indicators of self-concept - how well students feel they do in math (internal comparison) and how difficult math is for them compared to others (external social comparison). It also discusses how parental education, a factor influenced by population migration during the transition, relates to both self-concept and achievement and may have shown a non-monotonic trend over the time period examined. The study aims to investigate how parental education changes relate to self-concept and achievement development and examine whether the reciprocal relationship model between
El actor Luis Machín terminó de protagonizar la obra "La última sesión de Freud" junto a Jorge Suárez, la cual tuvo más de 300 funciones y recibió elogios de la industria. Machín habla sobre su trayectoria actoral de décadas, sus diversos roles protagónicos en cine, teatro y TV, y la experiencia de interpretar personajes históricos y políticos controvertidos.
Atendendo Milhares de Requisições com o Play Framework 2 - v2Paulo Siqueira
Com testes de carga, esta palestra mostra como a Lojinha (lojinha.jcranky.com e github.com/jcranky/lojinha) pode atender a milhares de requisições, sem complicações. A primeira versão desta palestra foi apresentada no Just Java 2013, e a segunda (atual) no TDC 2013 em SP.
Este documento presenta la Norma Internacional de Auditoría 805 sobre consideraciones especiales para auditorías de estados financieros únicos y elementos, cuentas o partidas específicas de un estado financiero. La norma establece requisitos para aceptar el trabajo, planificar y realizar la auditoría, y formar una opinión y consideraciones del dictamen. La norma se aplica a auditorías de información financiera histórica individual como estados financieros únicos o elementos específicos preparados de acuerdo con marcos de referencia de propósito general o especial
La contabilidad financiera es la técnica de registrar, clasificar, sumarizar y presentar de manera sistemática y cuantitativa las transacciones económicas de una entidad para informar a terceros sobre su situación financiera y resultados a través de estados financieros. Tiene como objetivo principal rendir cuentas a inversionistas, acreedores y otros grupos de interés externos.
This document outlines a 16-session training course on IBM WebSphere MQ. The sessions cover MQ architecture, messaging concepts, objects like queues and channels, distributed queue management, triggering, commands, dead letter queues, utilities, installation, configuration files, clusters, multi-instance usage, and protocols. Hands-on exercises are included to demonstrate topics like one-way and two-way communication, triggering, dead letter queue processing, and how clusters work.
Los sistemas distribuidos se definen como sistemas cuyos componentes de hardware y software se comunican a través de una red para lograr un objetivo común. Estos sistemas se caracterizan por la compartición de recursos entre máquinas, su apertura para agregar nuevos servicios, y su capacidad para ejecutar procesos de forma concurrente y escalable. Los sistemas distribuidos evolucionaron desde modelos centralizados hacia arquitecturas cliente-servidor más descentralizadas para mejorar el rendimiento y tolerancia a fallos. Existen diferentes clasificaciones
James Perkins is an animator and rigger who is passionate about using his technical skills and problem-solving abilities to inspire and lead others. He grew up enjoying art and animation from a young age and developed a desire to pursue a career where he could combine his artistic talents with his love of technology. His goal is to improve animation programs and share his work and insights through his art blog to motivate and help other artists.
O documento discute a economia do mar em Portugal, destacando que a aquicultura representa uma grande oportunidade de mercado, com potencial para 300 mil toneladas anualmente, ajudando a reduzir a dependência das importações de pescado. A aquicultura é citada como a maior oportunidade de investimento do século 21.
На вебинаре вы узнаете:
(!) О том, как планируются серьезные Agile проекты
(!) О том, кто учавствует в планировании
(!) Философские взгляды на Agile-планирование
Este documento presenta varios casos de gripe A (H1N1) en diferentes países y regiones. En Argentina, 7 personas murieron de gripe A (H1N1) en la provincia de Buenos Aires. En Bogotá, Colombia, 3 personas murieron recientemente del virus H1N1. En Colombia también se reportaron 4 muertes por gripe A (H1N1) en el departamento de Norte de Santander. Finalmente, una adolescente de 15 años en el estado de Falcón, Venezuela fue la primera víctima reportada del virus H1N1 en esa región.
Este documento presenta la Norma Internacional de Auditoría 810 sobre los trabajos para dictaminar estados financieros resumidos. La NIA establece los requisitos para que un auditor acepte y realice trabajos para emitir opiniones sobre estados financieros resumidos derivados de estados financieros auditados. La NIA especifica los procedimientos que el auditor debe llevar a cabo y los elementos que debe contener el dictamen del auditor sobre los estados financieros resumidos.
The document discusses potential solutions. It suggests exploring alternative approaches and considering additional factors. Overall, it focuses on finding an effective and balanced resolution.
QQ is an instant messaging software launched in 1999 that allows users to communicate via text, voice, and video chat. It has over 829 million active accounts. Users can find friends by searching for their unique QQ IDs, and can then chat, share files, use QQ music, and write emails. QQ also offers additional services like Qzone for blogging, QQ Doctor for tech support, and Tencent Web Mini for web access. While QQ is commonly used on computers and laptops, WeChat is more popular for mobile messaging due to its private chat settings and use of 'moments' like a social feed.
SMP26 Realidad Aumentada para campañas de comunicación y marketingSocialMediaPoint
El documento describe una campaña de marketing para Cheetos que involucra tatuajes mágicos en las bolsas que pueden escanearse con una aplicación para recolectar animaciones. La aplicación Cheetos Moovins obtuvo más de 400,000 descargas y 3 millones de visualizaciones en 10 semanas. La campaña se implementó en España, Portugal y Grecia a través de televisión, empaques, posters y más para involucrar a los consumidores de la Generación Z de una manera digital.
Este documento apresenta constantes físico-químicas e definições de termos utilizados em química, como a constante de Avogadro, constante de Faraday, volume molar de gás ideal, entre outras. Além disso, fornece massas molares de alguns elementos químicos.
La Unión Europea ha acordado un paquete de sanciones contra Rusia por su invasión de Ucrania. Las sanciones incluyen restricciones a las importaciones de productos rusos de alta tecnología y a las exportaciones de bienes de lujo a Rusia. Además, se congelarán los activos de varios oligarcas rusos y se prohibirá el acceso de los bancos rusos a los mercados financieros de la UE.
This document discusses organizational culture and motivation. It introduces how organizational culture was initially studied in the 1970s and 1980s, focusing on how culture could provide competitive advantages. It describes companies as complex, unstable, interdependent systems where culture is co-constructed through daily interactions. The key to a strong culture is focusing on motivating employees through play, purpose, or allowing them to reach their potential, rather than relying on economic or emotional pressures. A successful example is given of Southwest Airlines, which combines purpose, play, and potential to motivate its employees. The conclusion states that focusing on enhancing culture through consistent behaviors, rather than just discussing it, is important for companies.
Algebraic Thinking A Problem Solving ApproachCheryl Brown
This document discusses developing algebraic thinking through a problem solving approach in mathematics education. It argues that solving mathematical problems can help students develop a deeper understanding of algebraic concepts by encouraging them to consider the generalities and relationships within problems. The document outlines several key aspects of algebraic thinking, such as reasoning about patterns, generalizing, and thinking about mathematical relations. It advocates facilitating student discourse and communication of mathematical ideas to promote algebraic reasoning. Finally, it suggests students progress through stages of developing algebraic thinking, starting with describing generalities verbally and moving to using diagrams, symbols, and formal algebraic notation.
This document outlines the goals, framework, and structure of the revised Grades 1-10 Mathematics curriculum in the Philippines. The main goal is for learners to become mathematically proficient and critical problem solvers. The curriculum framework focuses on three facilitating facets - content, skills, and disposition - supported by pedagogy, assessment, and resources. The curriculum structure is built around 12 "Big Ideas" that connect mathematical concepts across grade levels through a developmental sequence and vertical/horizontal articulation.
El actor Luis Machín terminó de protagonizar la obra "La última sesión de Freud" junto a Jorge Suárez, la cual tuvo más de 300 funciones y recibió elogios de la industria. Machín habla sobre su trayectoria actoral de décadas, sus diversos roles protagónicos en cine, teatro y TV, y la experiencia de interpretar personajes históricos y políticos controvertidos.
Atendendo Milhares de Requisições com o Play Framework 2 - v2Paulo Siqueira
Com testes de carga, esta palestra mostra como a Lojinha (lojinha.jcranky.com e github.com/jcranky/lojinha) pode atender a milhares de requisições, sem complicações. A primeira versão desta palestra foi apresentada no Just Java 2013, e a segunda (atual) no TDC 2013 em SP.
Este documento presenta la Norma Internacional de Auditoría 805 sobre consideraciones especiales para auditorías de estados financieros únicos y elementos, cuentas o partidas específicas de un estado financiero. La norma establece requisitos para aceptar el trabajo, planificar y realizar la auditoría, y formar una opinión y consideraciones del dictamen. La norma se aplica a auditorías de información financiera histórica individual como estados financieros únicos o elementos específicos preparados de acuerdo con marcos de referencia de propósito general o especial
La contabilidad financiera es la técnica de registrar, clasificar, sumarizar y presentar de manera sistemática y cuantitativa las transacciones económicas de una entidad para informar a terceros sobre su situación financiera y resultados a través de estados financieros. Tiene como objetivo principal rendir cuentas a inversionistas, acreedores y otros grupos de interés externos.
This document outlines a 16-session training course on IBM WebSphere MQ. The sessions cover MQ architecture, messaging concepts, objects like queues and channels, distributed queue management, triggering, commands, dead letter queues, utilities, installation, configuration files, clusters, multi-instance usage, and protocols. Hands-on exercises are included to demonstrate topics like one-way and two-way communication, triggering, dead letter queue processing, and how clusters work.
Los sistemas distribuidos se definen como sistemas cuyos componentes de hardware y software se comunican a través de una red para lograr un objetivo común. Estos sistemas se caracterizan por la compartición de recursos entre máquinas, su apertura para agregar nuevos servicios, y su capacidad para ejecutar procesos de forma concurrente y escalable. Los sistemas distribuidos evolucionaron desde modelos centralizados hacia arquitecturas cliente-servidor más descentralizadas para mejorar el rendimiento y tolerancia a fallos. Existen diferentes clasificaciones
James Perkins is an animator and rigger who is passionate about using his technical skills and problem-solving abilities to inspire and lead others. He grew up enjoying art and animation from a young age and developed a desire to pursue a career where he could combine his artistic talents with his love of technology. His goal is to improve animation programs and share his work and insights through his art blog to motivate and help other artists.
O documento discute a economia do mar em Portugal, destacando que a aquicultura representa uma grande oportunidade de mercado, com potencial para 300 mil toneladas anualmente, ajudando a reduzir a dependência das importações de pescado. A aquicultura é citada como a maior oportunidade de investimento do século 21.
На вебинаре вы узнаете:
(!) О том, как планируются серьезные Agile проекты
(!) О том, кто учавствует в планировании
(!) Философские взгляды на Agile-планирование
Este documento presenta varios casos de gripe A (H1N1) en diferentes países y regiones. En Argentina, 7 personas murieron de gripe A (H1N1) en la provincia de Buenos Aires. En Bogotá, Colombia, 3 personas murieron recientemente del virus H1N1. En Colombia también se reportaron 4 muertes por gripe A (H1N1) en el departamento de Norte de Santander. Finalmente, una adolescente de 15 años en el estado de Falcón, Venezuela fue la primera víctima reportada del virus H1N1 en esa región.
Este documento presenta la Norma Internacional de Auditoría 810 sobre los trabajos para dictaminar estados financieros resumidos. La NIA establece los requisitos para que un auditor acepte y realice trabajos para emitir opiniones sobre estados financieros resumidos derivados de estados financieros auditados. La NIA especifica los procedimientos que el auditor debe llevar a cabo y los elementos que debe contener el dictamen del auditor sobre los estados financieros resumidos.
The document discusses potential solutions. It suggests exploring alternative approaches and considering additional factors. Overall, it focuses on finding an effective and balanced resolution.
QQ is an instant messaging software launched in 1999 that allows users to communicate via text, voice, and video chat. It has over 829 million active accounts. Users can find friends by searching for their unique QQ IDs, and can then chat, share files, use QQ music, and write emails. QQ also offers additional services like Qzone for blogging, QQ Doctor for tech support, and Tencent Web Mini for web access. While QQ is commonly used on computers and laptops, WeChat is more popular for mobile messaging due to its private chat settings and use of 'moments' like a social feed.
SMP26 Realidad Aumentada para campañas de comunicación y marketingSocialMediaPoint
El documento describe una campaña de marketing para Cheetos que involucra tatuajes mágicos en las bolsas que pueden escanearse con una aplicación para recolectar animaciones. La aplicación Cheetos Moovins obtuvo más de 400,000 descargas y 3 millones de visualizaciones en 10 semanas. La campaña se implementó en España, Portugal y Grecia a través de televisión, empaques, posters y más para involucrar a los consumidores de la Generación Z de una manera digital.
Este documento apresenta constantes físico-químicas e definições de termos utilizados em química, como a constante de Avogadro, constante de Faraday, volume molar de gás ideal, entre outras. Além disso, fornece massas molares de alguns elementos químicos.
La Unión Europea ha acordado un paquete de sanciones contra Rusia por su invasión de Ucrania. Las sanciones incluyen restricciones a las importaciones de productos rusos de alta tecnología y a las exportaciones de bienes de lujo a Rusia. Además, se congelarán los activos de varios oligarcas rusos y se prohibirá el acceso de los bancos rusos a los mercados financieros de la UE.
This document discusses organizational culture and motivation. It introduces how organizational culture was initially studied in the 1970s and 1980s, focusing on how culture could provide competitive advantages. It describes companies as complex, unstable, interdependent systems where culture is co-constructed through daily interactions. The key to a strong culture is focusing on motivating employees through play, purpose, or allowing them to reach their potential, rather than relying on economic or emotional pressures. A successful example is given of Southwest Airlines, which combines purpose, play, and potential to motivate its employees. The conclusion states that focusing on enhancing culture through consistent behaviors, rather than just discussing it, is important for companies.
Algebraic Thinking A Problem Solving ApproachCheryl Brown
This document discusses developing algebraic thinking through a problem solving approach in mathematics education. It argues that solving mathematical problems can help students develop a deeper understanding of algebraic concepts by encouraging them to consider the generalities and relationships within problems. The document outlines several key aspects of algebraic thinking, such as reasoning about patterns, generalizing, and thinking about mathematical relations. It advocates facilitating student discourse and communication of mathematical ideas to promote algebraic reasoning. Finally, it suggests students progress through stages of developing algebraic thinking, starting with describing generalities verbally and moving to using diagrams, symbols, and formal algebraic notation.
This document outlines the goals, framework, and structure of the revised Grades 1-10 Mathematics curriculum in the Philippines. The main goal is for learners to become mathematically proficient and critical problem solvers. The curriculum framework focuses on three facilitating facets - content, skills, and disposition - supported by pedagogy, assessment, and resources. The curriculum structure is built around 12 "Big Ideas" that connect mathematical concepts across grade levels through a developmental sequence and vertical/horizontal articulation.
The document compares and contrasts didactics of mathematics and mathematics education through comparative charts provided by multiple authors. Didactics of mathematics focuses on identifying and understanding phenomena related to teaching and learning mathematics, while mathematics education includes the broader field of teaching, learning, and researching mathematics. Both fields aim to improve the learning and understanding of mathematical concepts but do so through different approaches and areas of focus.
Authentic Tasks And Mathematical Problem SolvingJim Webb
This document discusses authentic tasks in mathematical problem solving and their role in developing mathematical literacy. It describes four key dimensions of authentic tasks: thinking and reasoning, discourse, mathematical tools, and attitudes and dispositions. Each of these dimensions supports meaningful learning and prepares students to solve everyday problems. The document provides examples of lessons and programs that incorporate these dimensions through real-world, problem-based activities.
This document discusses pedagogy for transferring mathematics learning from school to the workplace. It begins by defining numeracy and noting how definitions emphasize understanding mathematics in real-world contexts and as a tool for communication. The document then reviews different views of the nature of mathematics and how these influence teaching practices. It argues that developing numerate learners requires shifting curricula from views of mathematics as facts and skills to seeing it as problem-solving arising from human inquiry. A second theme is functional mathematics, which aims to bridge the gap between school and out-of-school mathematics by focusing on areas inherent to employment. The document aims to examine how curriculum and teaching can better serve the needs of students and other stakeholders in transferring meaningful mathematics learning
Dr. Anthony Rickard, www.nationalforum.com
In June 2008, Dr. Kritsonis received the Doctor of Humane Letters, School of Graduate Studies from Southern Christian University. The ceremony was held at the Hilton Hotel in New Orleans, Louisiana.
Proposal Submitted to John Dewey Academy in Great Barrington, MAJonathan Dunnemann
This document proposes establishing the John Dewey Academy pilot study group and outlines dimensions of learning based on John Dewey's theories of education. It discusses five dimensions: 1) fostering positive attitudes about learning, 2) acquiring and integrating knowledge, 3) extending and refining knowledge through thinking, 4) using knowledge in meaningful tasks, and 5) developing habits of mind. The document provides background on Dewey's views of reflective thinking and explains how the dimensions parallel his concepts.
4. Statistical Thinking In A Technological EnvironmentRenee Lewis
The document discusses a statistics curriculum developed for junior high school students in Israel that incorporates technology. It aims to make statistics learning more meaningful by having students actively solve problems using real data. The curriculum includes both structured activities where students investigate open-ended problems through the statistical investigation process, as well as open-ended research projects done in small groups. Technology allows students to more easily collect, analyze, and visualize large real-world data sets when investigating problems.
The document discusses how ICT can support collaborative creativity in modern schools. It makes three key points:
1. ICT tools like videoconferencing and web 2.0 applications allow students to collaboratively create, share, and build knowledge from remote locations in real time. This encourages a participatory culture of sharing ideas.
2. The ODYSSEUS program used ICT tools to have primary students from different schools collaboratively design and implement creative activities together remotely. Four stages of the process are outlined.
3. ICT supports collaborative creativity by enabling problem solving, sharing creative works, and social interaction between students. This helps students develop important skills for the knowledge society.
This document analyzes a 6th grade mathematics module that incorporates Yup'ik culture and addresses goals of multicultural education. The module teaches concepts of proof, properties, perimeter, and area through having students model and analyze the construction of traditional Yup'ik fish racks. Analyzing the module shows how it connects mathematics learning to an authentic cultural activity, allowing Alaska Native students to learn in a culturally meaningful way while meeting state standards. Research found such culturally-based modules improved mathematics achievement for all students, especially Alaska Natives.
Mathematical literacy involves more than procedures - it requires a knowledge base and confidence to apply math skills to practical problems. A mathematically literate person can estimate, interpret data, solve daily problems, and communicate using math. As technology and the economy evolve, more jobs require math skills like problem solving and processing information. Math is also needed for everyday life outside of work. Mathematical literacy is key to coping with societal changes and is as important as reading and writing literacy. Understanding basic math concepts is necessary to comprehend information in today's complex world and opens doors to opportunities.
This document summarizes key points from a presentation about communication design and theories of learning. It discusses how communication designers produce information to explain technologies and work in ill-structured problem domains. It also outlines general problem-solving tasks and what is known about learning, including comprehension, representation, retrieval, and construction of new understandings.
Rickard, anthony analysis of a culturally based sixth grade mathematics modul...William Kritsonis
This document analyzes a 6th grade mathematics module called "Building a Fish Rack" that is part of a larger culturally-based mathematics curriculum. The module uses the cultural practice of constructing fish racks among the Yup'ik people of Alaska as a context for teaching mathematical concepts of perimeter, area, and geometry. The analysis discusses how the module addresses goals of multicultural education by (1) raising mathematics achievement for all students, (2) providing access to rigorous mathematics content, and (3) using an ethnomathematical approach that engages students through a culturally relevant context. Research on the module shows improved mathematics outcomes for students compared to a traditional curriculum.
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An overview of the field of education
1. AN OVERVIEW OF THE FIELD OF EDUCATION
Diciplinary Perspective
The context of inquiry is made up of a complex web in which our underlying beliefs carried
out within the local dynamics of particular investigation. Not only is the individual situated
within a given research setting, he or she is also situated within a particular epistemological,
cultural, and gendered framework of beliefs and values that both facilitate and constrain how
we perceive the world and what we select for study within.
Eisner (1993), using historical perspective, illustrates how particular perceptions can
influence the framing and examination of educational practice.
How we answer the question of whether history is the text historians write or the past
historians write about is crucial to our own view of what history is and, therefore, to what is
relevant for helping students understand it. If history is text, then text must continue to be
central to the teaching of history: To understand history one has to understand text. But if
history is the past about which historians write, then any form of representation that sheds
light on the past is relevant, indeed a useful, way to understand history. (P.9).
Mathematics educators face a similar problem in considering their field of inquiry.
Underlying assumtions concerning the nature and role of mathematics in society and how it is
to be taught and learned influence both educational practice and research (Dossey,1992). A
recognition of the changing face of 20th century mathematics, the emergence of computers
and powerful handheld calculator, and the changing needs of contemporary sociey raise
questions about the appropriateness of specific educational content. Emerging theories of
learning suggest new approaches to traditional classroom and question old assumptions about
the kinds of mathematical knowledge that best promote understanding (R.B Davis, 1994).
One role of research in mathematics education is to illuminate these issues and provide an
information base for their resolution ( Research Advisory Committee, 1997). A great deal has
been written about these issues (Davis: Dossey,1992). The following discussion represents
only the tip important disciplinary iceberg that bears careful scrutiny elsewhere.
The nature of Mathematics
Mathematics has been described as the science of abstract patterns and characterized by its
usefulness for organizing mental and empirical structures (Devlin....1996). This
Characterization only hints at the complex nature of a field that is valued by some for its
intellectual beauty, by others for its utilitarian applications, and by still others for its
emancipatory properties in an increasingly technological world. Mathematics can be regarded
as the product of intellectual abstaction or as the processes that produce such a product. It can
be regarded as a static entity or as a fallible, creative, evolving activity that permeates many
aspects of our daily lives.
The subject of mathematics is multifaceted. Steen (1990) describes a set of diverse
perspectives that “illustrate the complexity of structures that support mathematics”(p.4).
These perspectives, or “ deep ideas that nourish the growing branches of mathematics”(p.3),
2. can be thougth of as (a) specific mathematical structures, such as number or shapes ; (b)
mathematical attributes, such as linear or periodic;(c) actions, such as represent or prove;(d)
abtractions, such as symbols or equivalence;(e) attitudes, such as wonder or beauty; (f)
mathematical behaviors, such as motion or iteration; or (g) mathematical dichotomies, such
as discrete versus continuous.
Another characteristic of the subject is that many mathematical entities exhibit a
process/product duality reflecting an “interplay of form with content”
(Freudenthal,1991,p.10). Symbolic representations of such entities can be perceived either as
mathematical processes or as the products of these processes. Mathematical thought is
characterized by the ability to generalize detail (process or from) into structure (product or
content) and to create new conceptual entities from an abstraction of this structure
(Sfard,1991;....1995). This movement from form to content makes it possible t deal with
complexity by reducing detail through abstaction (Devlin....1991). In addition, the use of
some form of external representation makes it possible think about and manipulate the
concepts under consideration, which leads to new levels of thought. The search for, and
development of, adequate symbols to express new levels of conceptualization are a
characteristic of the generative power of mathematics (van Oers, 1996).
The nature of mathematics is also reflected through the activity and disposition of its
practitioners. Doing mathematics, which is different from the mathematics presented in
professional journals, includes, among other actions, exploring situations, searching for
patterns. Inventing strategies, using intuition, generalizing, and abstracting-creating
mathematics. Coupled with such activity is the development of a mathematical disposition-a
particular way of perceiving and doing that enables a person to navigate his or her way
around the domain. Breiter (1997) describes this as “ a diffuse kind of knowledge or
competence that makes performance possible”(p.3). Cobb and Yackel (1996) identify a set of
reflexively related beliefs, values, and “sociomathematical norms” that, developed through
classroom interactions, enables students to engage in autonomous mathematical practice.
Mathematics exists as an aim in itself of concern to practicing mathematicians, as a tool
applied in the pursuit of other disciplines or in everyday life, and as a social and cultural
artifact. As an aim in itself, it is self-generating, serving as “a forceful motor for its own long-
term development”(Freudenthel....3). Within this area, mathematical objects are created. As a
tool, it is used to mathematics (P.J.Davis,1992). When employed in the course of everyday
activities, such as in bulding a house or selling goods, the term etnomathematics has been
developed to emphasize the cultural embeddedness of the tool (Nunes,1992).
Our understanding of mathematics is grounded in prectice that is grounded in practice that is
implicitly social. In his sociology of mathematics, Restivo (1992) argues that mathematical
objects do not exist “independent of the flux of history and....embody world views”(p.103). It
is necessary to think in term of a “cultural conception of mathematics”(Radford,1997,p.28).
Mathematics is also grounded in common sense and stems from the systematization,
organization, and reorganization of contextually bound experiences (Freudenthal,1991).”The
3. effects of culture and society are fundamental to the way in which we come to know”
(Radford,1997,p.29).
The Nature of Mathematics Education
It is important for researchers engaged in the processes of “giving meaning to educational
events” to situate these meanings against one’s “assumptions about the teaching and
schooling that underline those events” (Gitlin, 1990, p.460). How one think about
mathematics education, how one defines what it means for students to know and do
mathematics in school, is, in turn, affected by one’s view about the nature of mathematics,
one’s underlying epistemological perspective, and one’s educational goals.
P.J.Davis (1993) argues that today’s world is characterized by the large degree to which
mathematizations permeate our daily lives, both in the humanistic areas and in the sciences.
Everyone uses mathematics at some level. Consequently, it is the role of education to enable
citizens to become aware of and assess these mathematizations-to develop”mathematical
estreet smarts’ that enable (them) to from judgments in the absence of technical
expertise”(p.192)
Breiter (1997) addresses the role of education from a more indivudual perspective,
advocating the development of mathematical disposition-an “Intuitive, perception-like
understanding...that makes lifelong learning in mathematics a possibility”(pp,3,5). Renz
(1997) offers a similar vision foe mathematics education, starting a minimal list of
competencies required for an uncertain future. Student should know how to solve problems
by asking others; Should be able to communicate by speaking, writing, and drawing; should
be aware of the existence of multiple solutions; and should understand that most problems do
not have definte solutions. Cobb and Yackel (1996) introduce additional dispositional
attributes from their social constructivist perspektive. In particular, mathematics education
should foster the development of sociomathematical norms including the development of
student autonomy and the ability to judge mathematical solutions on the basis of their
differences, sophistication, efficiency and acceptability. A common theme running through
the different educational criteria listed here is that what students believe and think about
mathematics is important for succeeding both in and out of school and for facilitating future
learning.
Mathematics education can be defined as formal schooling to distinguish it from the
ethomathematics of everyday learning.The operationalization of such education is then to
take mathematical knowledge. The operationalization of such education is then to take
mathematical knowledge which was originally developed to be use rather than taught, and
tranform it into to teachable form (Greer,1996) This tranformation proses must concider not
only what knowledge is to be learned, but also the nature of the knowledge and the kinds of
experiences that students are to develop (R.B. Davis,1994). The fokus of the transformational
process is on development of appropriate eductional tasks within effective learning
environments.
4. The Netherlands has developed a program of realistic mathematics education that employs
the processes of horizontal and vertical mathematicizing as the “teachable form”
(frudenthai......,1991). Conceptual development proceeds from informal, contect-bound-
experiences to mathematical for malisms. A given situatuon is horizontally mathematized by
students into a model; trough vertical mathematization, this model is transformed, again by
students, into formal mathematica structure. The emphasis is realistic mathematics education
is to use contextual situations that connect with children’s existing methds of working and
that promote natural, further generalizations and abstractions. Education is seen as the
process of guided reinvention in which the learner reinvents “mathematising rather than
mathematics; abstracting rather than abstractions;.....algorithmising rather than algorithms;
verbalising rather than languange”(Freudenthal,1991,p.49).
Learning mathematics is recognized as a social and cultural activity. Our schools serve as one
of pleces in which students are introduced to the “meaning of culturally approved
mathematical signs, symbols, and techniques” (Crawford,1996,p.145). The question of
wether this process is enculturation (the assimilation of an existing tradition) or acculturation
(the process of inter-cultural borrowing to create a new and blended culture) is one of
perspective. When a mathematics classroom examined from the external vantage point of an
educational system, that is, outside the culture of the classroom, the processes of educational
system, that is, outside the culture of the classroom, the processes of education are seen as the
enculturation of students as they interact with more knowledgeable others. From the point of
view within the culture of the clasroom, the differences in beliefs and values of the
participants support a view of accultoration in which the intersubjective meanings of the
studentsand teacher are negotiated in the process of constituting a classroom mathematics
community (Cobb, Jaworski.......1996). Learning environment can be thought of has being
constructed by individuals in activity rather than as existing independently of the participants
(Saxe & Bermudez, 1996)
Richards (1996) describes a type of mathematics classroom that fosters acculturation into a
shared community through the use of what he calls,”inquiry math”. In such a calssroom, the
students and teacher participate in mathematical discussion and act mathematically-asking
questions, solving problems that are problematic to the solvers, posing conjectures, and
listening to mathematical arguments. The classroom atmosphere allows taking risks and
making mistakes, and the teacher is able to truly listen to the students because of his or her
understanding of “the large mathematical picture that provides a context for the students’
questions" (p.74). Mathematical communication and the negotiation of meaning take place at
a level at which there are two sides, and each is able to listen to the other.
The preceding characterizations illustrate the diversity of perspectives from which school
mathematics can be viewed. Each perspective highlights a different aspect of the complex
reality of mathematics learning within a socially and culturally constituted environment. As
R.B Davis (1994) points out in his discussion of what mathematics students should
learn,”people no more agree on what they most value in an act of mathematical problem
solving than they do in paintings or poems or symphonies. I know people who can’t see why
anyone would want to listen to Bach”(p.25).
5. Mathematics Education research
The multiple perspectives from which the nature of mathematics is now being considered and
the variations in processes and contexts that are increasingly being used to characterize
school mathematics reflect a paradigm shift from a modernist to a postmodernist worldview.
This shift represents a reconceptualization of the nature of knowledge from a single and
external reality to a set of multiple and subjective realities. Current trends in research in
mathematics education reflect a similar peradigm shift from an emphasis on scientific or
quantitative studies to the use of qualitative, interpretative methodologies (see Ernest,
Chapter 3, this volume).”Researches today are looking at aspect of mathematical learning in
ways that were, if not unthought of, at least not common 25 years ago”(Kieran,1994,p583)
The goals of mathematics education reserch reflect the diversity and complexity of its
subjects of inquiry. At a general level the goal of research is to produce new knowledge. In a
field that is still in the process of defining itself, this knowledge provides an important base
for progress in mathematics education encompasses the study and implementation of
classroom intstruction (in all its complexity) as well as the formulation of theories of the
development of mathematical understanding (in all its variations).
From a practical perspective, there is a need for research to address the concerns of practicing
educators in the field and to present results that are accessible to them and are of immediate
use (Kennedy,1997). The questions and data from such studies also provide important
examples of exemplary practice and resources to stimulate reflection on relevant issues
(Research Advisory Committe.....1997). Research is also needed to facilitate the creation
implementation, and evaluation of innovative curricula that can move mathematics programs
beyond simply making improvements to those that currenty exist. In addition, researchers
should consider what role they can, or should, play in the development of informed public
discussion concerning educational progress (R.B.Davis........1997).
Research on practice can also be used as an emancipatory vehicle. The processes of inquiry,
when conducted by practicing teachers, give them voice and power and enable them to
pursue lines of inquiry directly applicable to their needs. The teacher-researcher and
providing them with a vechile for effecting change and setting educational policy
(Brown......1994)
The incorporation of qualitative methodologies into mathematics education researh has made
it possible to investigate the teaching and learning of mathematics at new and different levels
of complexity and from multiple perspectives. A goal of much of this research is to
investigate the processes of coming ti know mathematics both from the perspective of the
cognizing individual and from within the sociocultural interactions of the classroom. Such
research aids in the development of explanatory models of what cnstitutes mathematical
learning (Steffe,1996)
Research has also been used to expand our own conceptions of mathematics and our
perceptions of what is possible in mathematics education. Attending to students’ expressiona
of their ways of thinking, doing, and describing, or “student voice” promotes
6. reconceptualizations of one’s own mathematical uderstanding and promotes diversity of
mathematical perspective (Confrey,1995). Careful analysis of videotapes of groups of
students working through problematic situations reveals students’ abilities to develop
powerful mathematical ways of thinking (Maher&Martino 1996).
Multiple Challenges face the field of mathematics education research. The availability of
diverse perspectives for launching inquiry will require a greater acceptance of alternative
research methodologies and greater commitment for researchers to communicate with one
another across divergent points of view. As we recognize and accept the complexity the
questions we wish to ask and the phenomenon we wish to study, new methods will need to be
identified that can capture this complexity in meaningful ways-requiring a “new set of
explanations and a new set tools”
The dynamic nature of the field of mathematics educatioan research is reflected an interplay
between emerging lines of inquiry and evolving research methods. At the same time, the
growth in diversity of perspectives makes it more difficult to establish standards of scholarly
inquiry on which all participants in the field can agree. As silver and Kilpatrick (1994) point
out.
It may be neither possible nor desirable to forge a single perspective out of the many that are
found in our field...[What is needed is a] spirit of greather openness, tolerance, and respect
for the work and ideas of those colleagues who share neither our culture nor our
tradition.(p763)