No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
Pre-Service and In-Service Mathematics Teachers’ Knowledge and Professional Development
Abstract
As Adler, Ball, Krainer, Lin and Novotna (2005) remarked in their landmark ICME survey report, research into mathematics teacher education was rather sparse until the mid-1990s. From its roots in mathematics and psychology – witness the name of the sponsor organisation of this handbook – the output of researchers in mathematics (or ‘mathematical’) education had previously been more directed, and often in an anecdotal way, towards learners, curricula, purposes and innovative instruction (Kilpatrick, 1992).
... Research on PS teaching has generated a need both to expand on certain particulars and to reinterpret teachers' knowledge models, which are overly general and tend to omit some issues related to PS (Lin & Rowland, 2016). Our study builds on the research work by Foster et al. (2014) and Chapman (2015). ...
... Our study builds on the research work by Foster et al. (2014) and Chapman (2015). Foster et al. (2014) voice a conservative caveat about Ball's research (Lin & Rowland 2016). Their re-write of Ball et al.'s (2008) model intensifies the difficulty, highlighted by Carrillo et al. (2013), of differentiating between common and specialised knowledge of concepts and processes. ...
... En los últimos años, diversos investigadores han evidenciado que los modelos de conocimiento del profesor son demasiado generales, omitiendo algunos aspectos en la configuración del conocimiento (Lin y Rowland, 2016). Burns y Lash (1988) examinan cómo las concepciones sobre la enseñanza de la matemática influencian la forma en que se planifica la instrucción de la resolución de problemas. ...
La resolución de problemas no solo es fundamental para aprender y hacer matemáticas, sino que también se considera una de las competencias necesarias para enfrentarse a los desafíos de las sociedades actuales. Ayudar a los estudiantes a convertirse en resolutores de problemas competentes en matemáticas les proporciona una forma de pensar para interactuar con problemas de la vida diaria. Para lograr esto, los profesores deben tener conocimientos especializados para enseñar la resolución de problemas matemáticos (Chapman, 2015; Foster, Wake y Swan, 2014). Por lo tanto, este conocimiento es una parte esencial del conocimiento que los futuros maestros necesitan desarrollar en sus programas de formación docente.
Una investigación que explore hasta qué punto los futuros maestros desarrollan este conocimiento es importante para mejorar los programas de formación y proporcionarles oportunidades de aprendizaje adecuadas. Dicha investigación requiere técnicas enfocadas específicamente en la perspectiva de la resolución de problemas como proceso (Foster et al., 2014). Esto permitiría abordar las limitaciones en los modelos destinados a determinar la competencia profesional de los docentes (Weber y Leikin, 2016). Esta tesis doctoral se centra en una reflexión sobre este conocimiento y lo concretiza en el conocimiento profesional manifestado por un grupo de estudiantes universitarios que terminan el Grado de Educación Primaria en la Universidad de Granada, acerca de la resolución de problemas en matemáticas.
El diseño metodológico de esta investigación se corresponde con un Diseño Mixto Exploratorio Secuencial (Creswell, 2013). Entendemos que los resultados de cada uno de los estudios producen un todo a través de la integración que es mayor que la suma de las partes cualitativas y cuantitativas individuales (Buchholtz, 2019).
En primer lugar, realizamos un estudio curricular. En este, utilizamos el marco de Chapman (2015) para analizar las pautas curriculares de seis países con resultados extremos en la evaluación PISA 2012 (Argentina, Chile, España, Estados Unidos, Finlandia y Singapur). El objetivo fue identificar los conocimientos necesarios para enseñar la resolución de problemas. Esto dio lugar a modificaciones en dicho marco, particularmente reacomodaciones y explicitaciones. Para realizar estas modificaciones en el conocimiento del contenido, utilizamos teorías de competencia matemática (e.g. Kilpatrick, Swafford y Findell, 2001) y teorías sobre competencia para resolver problemas (e.g. Chapman, 2015). En el caso del conocimiento pedagógico, nos valemos del triángulo didáctico (Schoenfeld, 2012), es decir, la relación entre profesor, alumno y contenido.
Esta primera fase ha permitido solventar en parte, las limitaciones que presentan los modelos de conocimiento del profesor (Foster et al., 2014). En esta caracterización identificamos tres elementos teóricos sobre los problemas y su resolución que deberían formar parte del conocimiento del profesor. Un primer elemento se relaciona con la noción de problema, un segundo con el proceso de resolver un problema y un tercer elemento con aspectos no cognitivos. Para el conocimiento didáctico, identificamos cuatro elementos teóricos, tres relativos al aprendizaje y uno a la enseñanza: (1) el estudiante como resolutor, (2) la resolución de problemas como tarea escolar, (3) factores no cognitivos que afectan la resolución de problemas, y (4) gestión de la enseñanza de la resolución de problemas.
Los componentes teóricos y el sistema de categorías refinado que provee el primer estudio ha permitido que en el segundo y tercer estudio, exploremos a través de encuestas, el conocimiento de futuros profesores sobre resolución de problemas en las matemáticas escolares. Particularmente, los conocimientos referidos al conocimiento del proceso y de su conocimiento pedagógico. Posteriormente, en base a los resultados obtenidos en el segundo estudio, en un tercer estudio profundizamos en dicho conocimiento del proceso a través de entrevistas. Este tercer estudio cualitativo estuvo centrado en los aspectos que presentaron conflictos (en la fase 2) entre los participantes.
Respecto a los resultados generales, señalar que el conocimiento que poseen los futuros maestros parece ser, mayoritariamente, de carácter teórico. Esto contrastaría con la competencia para resolver problemas exhibida por los futuros maestros al confrontar una actividad de resolución, particularmente de los futuros maestros españoles (e.g. Nortes y Nortes, 2016). Asimismo, los hallazgos sugieren que los futuros maestros poseen conocimientos pedagógicos sobre la resolución de problemas que no están organizados. Además, este conocimiento no refleja un aprendizaje reflexivo debido a que no son conscientes de las repercusiones que tienen ciertas acciones contra las que se declaran contrarias a la hora de preguntarlas de manera general. Por otra parte, un aspecto que la tercera fase puso de manifiesto es la desconexión entre sus conocimientos. Por ejemplo, los futuros maestros reconocen la importancia del resolutor y, por otro lado, reconocen la importancia de los conocimientos previos, pero no conectan estas ideas para establecer criterios en el etiquetado de tareas como problemas. Esto reafirma la importancia de la indagación y la reflexión en la formación inicial docente.
... Africa. Lin and Rowland (2016) were commissioned to review Psychology of Mathematics Education (PME) studies related to pre-service and in-service teachers' ...
... The review focused on several Mathematics teacher education issues including teacher knowledge, teacher beliefs, teacher education, educator education, professional development, and professional growth. Lin and Rowland (2016) found that there were two hundred and twenty papers submitted to PME with a focus on Mathematics teacher knowledge. ...
... The review by Jakobsen and Mosvold (2015) concur with Lin and Rowland's (2016) observation that there is limited research on MKT that has been conducted in Africa. ...
Previous studies showed that secondary school students fail to understand geometric proof development because they are not offered effective learning experiences. However, the studies do not describe the knowledge required by teachers to conduct effective geometric proving lessons. This study explored mathematical knowledge for teaching geometric proofs (MKT-GP) with an aim of providing insights into its content knowledge (CK) and pedagogical content knowledge (PCK). The categories of cognitive activation (COACTIV) model by Baumert and Kunter (2013) were used as overarching theory to inform the study in formulation of research questions, data analysis and presentation of findings. Qualitative case study design was utilised to generate and analyse data. Data were generated from four secondary school teachers through pencil and paper tests, individual interviews, and lesson observations. Both deductive and inductive thematic analyses were conducted on the data to provide CK and PCK for teaching geometric proving. The study has proposed the following categories of CK for geometric proof development: Geometry content knowledge, geometric reasoning, geometric deductive reasoning, problem solving skills and algebraic reasoning. The study has also proposed several sub-categories of PCK relevant for good assessment of students’ thinking, implementation of cognitively activating tasks, for explaining and representing geometric proofs. These include to knowledge of identifying causes of mistakes, knowledge of providing good guidance to students, knowledge of exploratory activities, knowledge of problem solving skills, and knowledge of good teaching and learning materials.
... Several papers have summarized the development of research on teachers and teaching in PME in the past (e.g., Hoyles 1992; da Ponte and Chapman 2006; Llinares and Krainer 2006;Jaworski 2011;Lin and Rowland 2016). In most contributions, the development of the field is described by three phases. ...
... In their summary of the field for the 30 years PME volume in 2006, da Ponte and Chapman (2006) identify four main objects of study in teacher-related research on PME: Teachers' mathematical knowledge, Teachers' knowledge of mathematics teaching, Teachers' beliefs and conceptions, Teachers' practices. While all of these topics are still in place, Lin and Rowland (2016) put teacher knowledge in the centre of their contribution for the 40 year handbook ten years later, highlighting its role in PME research. ...
... Advent of so-called "socio-cultural approaches": Several authors offer explanations for the increased focus on the teacher in the late 80s and in the 90s. Lin and Rowland (2016) note that this development coincided with the so-called "social turn", meaning an increased focus on social context, in which students' mathematical thinking and development takes place (see Sect. 13.3.4). Mathematical thinking and learning cannot be considered as something that happens in the students' isolated mind, independently of external influences. ...
Our work focuses on logic and language at a university in Cameroon. The mathematical discourse, carried by the language, generates ambiguities. At the university level, symbolism is introduced to clarify it. Because it is not taught in secondary school, it becomes a source of difficulties for students. Our thesis is as follows: “The determination of the logical structure of mathematical statements is necessary in order to properly use them in mathematics.” We conducted our study in the predicate calculus theory. In the first part of the paper, a summary of the theory is presented, followed by a logical analysis of two complex mathematical statements. The second part is a report of two sequences of an experiment that was conducted with first-year students that shows that knowledge of the logical structure of a statement enables students to clarify the ambiguities raised by language.
... Several papers have summarized the development of research on teachers and teaching in PME in the past (e.g., Hoyles 1992; da Ponte and Chapman 2006; Llinares and Krainer 2006;Jaworski 2011;Lin and Rowland 2016). In most contributions, the development of the field is described by three phases. ...
... In their summary of the field for the 30 years PME volume in 2006, da Ponte and Chapman (2006) identify four main objects of study in teacher-related research on PME: Teachers' mathematical knowledge, Teachers' knowledge of mathematics teaching, Teachers' beliefs and conceptions, Teachers' practices. While all of these topics are still in place, Lin and Rowland (2016) put teacher knowledge in the centre of their contribution for the 40 year handbook ten years later, highlighting its role in PME research. ...
... Advent of so-called "socio-cultural approaches": Several authors offer explanations for the increased focus on the teacher in the late 80s and in the 90s. Lin and Rowland (2016) note that this development coincided with the so-called "social turn", meaning an increased focus on social context, in which students' mathematical thinking and development takes place (see Sect. 13.3.4). Mathematical thinking and learning cannot be considered as something that happens in the students' isolated mind, independently of external influences. ...
This study examined how two middle school mathematics teachers changed from being reluctant to modify tasks in mathematics textbooks to having positive attitudes about textbook task modification. In order to successfully coordinate a curriculum revision with the textbooks they use, mathematics teachers need to be able to use their in-depth understanding of the intentions of both the revision and textbooks to modify and implement tasks appropriately. The two middle school teachers’ cases in this study showed that it is possible to change teachers’ negative attitudes about modifying tasks in mathematics textbooks if they explicitly understand the complexity in mathematics teaching and go through a sequence of activities that help them understand the revised curriculum in detail, interpret and modify textbook tasks, and implement the modified tasks and reflect on their implementation.
... Several papers have summarized the development of research on teachers and teaching in PME in the past (e.g., Hoyles 1992; da Ponte and Chapman 2006; Llinares and Krainer 2006;Jaworski 2011;Lin and Rowland 2016). In most contributions, the development of the field is described by three phases. ...
... In their summary of the field for the 30 years PME volume in 2006, da Ponte and Chapman (2006) identify four main objects of study in teacher-related research on PME: Teachers' mathematical knowledge, Teachers' knowledge of mathematics teaching, Teachers' beliefs and conceptions, Teachers' practices. While all of these topics are still in place, Lin and Rowland (2016) put teacher knowledge in the centre of their contribution for the 40 year handbook ten years later, highlighting its role in PME research. ...
... Advent of so-called "socio-cultural approaches": Several authors offer explanations for the increased focus on the teacher in the late 80s and in the 90s. Lin and Rowland (2016) note that this development coincided with the so-called "social turn", meaning an increased focus on social context, in which students' mathematical thinking and development takes place (see Sect. 13.3.4). Mathematical thinking and learning cannot be considered as something that happens in the students' isolated mind, independently of external influences. ...
The aim of this chapter is to give an overview of the research that we have been conducting in our research group in Mexico about the linear transformation concept, focusing on difficulties associated with its learning, intuitive mental models that students may develop in relation with it, an outline of a genetic decomposition that describes a possible way in which this concept can be constructed, problems that students may experience with regard to registers of representation, and the role that dynamic geometry environments might play in interpreting its effects. Preliminary results from an ongoing study about what it means to visualize the process of a linear transformation are reported. A literature review that directly relates to the content of this chapter as well as directions for future research and didactical suggestions are provided.
KeywordsLinear transformationVisualizationRepresentationDynamic geometryLinear algebra
... Several papers have summarized the development of research on teachers and teaching in PME in the past (e.g., Hoyles 1992; da Ponte and Chapman 2006; Llinares and Krainer 2006;Jaworski 2011;Lin and Rowland 2016). In most contributions, the development of the field is described by three phases. ...
... In their summary of the field for the 30 years PME volume in 2006, da Ponte and Chapman (2006) identify four main objects of study in teacher-related research on PME: Teachers' mathematical knowledge, Teachers' knowledge of mathematics teaching, Teachers' beliefs and conceptions, Teachers' practices. While all of these topics are still in place, Lin and Rowland (2016) put teacher knowledge in the centre of their contribution for the 40 year handbook ten years later, highlighting its role in PME research. ...
... Advent of so-called "socio-cultural approaches": Several authors offer explanations for the increased focus on the teacher in the late 80s and in the 90s. Lin and Rowland (2016) note that this development coincided with the so-called "social turn", meaning an increased focus on social context, in which students' mathematical thinking and development takes place (see Sect. 13.3.4). Mathematical thinking and learning cannot be considered as something that happens in the students' isolated mind, independently of external influences. ...
Algebra can be viewed as a language of mathematics; playing a major role for students’ opportunities to pursue many different types of education in a modern society. It may therefore seem obvious that algebra should play a major role in school mathematics. However, analyses based on data from several international large-scale studies have shown that there are great differences between countries when it comes to algebra; in some countries algebra plays a major role, while this is not the case in other countries. These differences have been shown consistent over time and at different levels in school. This paper points out and discusses how these differences may interfere with individual students’ rights and opportunities to pursue the education they want, and how this may interfere with the societies’ need to recruit people to a number of professions.
... Several papers have summarized the development of research on teachers and teaching in PME in the past (e.g., Hoyles 1992; da Ponte and Chapman 2006; Llinares and Krainer 2006;Jaworski 2011;Lin and Rowland 2016). In most contributions, the development of the field is described by three phases. ...
... In their summary of the field for the 30 years PME volume in 2006, da Ponte and Chapman (2006) identify four main objects of study in teacher-related research on PME: Teachers' mathematical knowledge, Teachers' knowledge of mathematics teaching, Teachers' beliefs and conceptions, Teachers' practices. While all of these topics are still in place, Lin and Rowland (2016) put teacher knowledge in the centre of their contribution for the 40 year handbook ten years later, highlighting its role in PME research. ...
... Advent of so-called "socio-cultural approaches": Several authors offer explanations for the increased focus on the teacher in the late 80s and in the 90s. Lin and Rowland (2016) note that this development coincided with the so-called "social turn", meaning an increased focus on social context, in which students' mathematical thinking and development takes place (see Sect. 13.3.4). Mathematical thinking and learning cannot be considered as something that happens in the students' isolated mind, independently of external influences. ...
This paper poses methodological questions concerning the evaluation of emotion in the process of mathematical learning where the interaction between emotion and cognition occurs. These methodological aspects are considered not only from the perspective of educational psychology but from that of mathematics education. Some epistemological and ontological aspects, which are considered central to the cognition-affect interplay, are noted. Special attention is given to the notion of cognitive-affective structure as a dynamic system. The interplay between cognition and affect in mathematics is viewed through the concepts of local and global affect and using a mathematical working space model. A model of this interplay is illustrated with research examples, enabling us to move from descriptions of cognition-affect at an individual level to the explanation of the tendency of a group. The non-linear modelling of emotion is reflected in the affect-cognition local structure.
... Several papers have summarized the development of research on teachers and teaching in PME in the past (e.g., Hoyles 1992; da Ponte and Chapman 2006; Llinares and Krainer 2006;Jaworski 2011;Lin and Rowland 2016). In most contributions, the development of the field is described by three phases. ...
... In their summary of the field for the 30 years PME volume in 2006, da Ponte and Chapman (2006) identify four main objects of study in teacher-related research on PME: Teachers' mathematical knowledge, Teachers' knowledge of mathematics teaching, Teachers' beliefs and conceptions, Teachers' practices. While all of these topics are still in place, Lin and Rowland (2016) put teacher knowledge in the centre of their contribution for the 40 year handbook ten years later, highlighting its role in PME research. ...
... Advent of so-called "socio-cultural approaches": Several authors offer explanations for the increased focus on the teacher in the late 80s and in the 90s. Lin and Rowland (2016) note that this development coincided with the so-called "social turn", meaning an increased focus on social context, in which students' mathematical thinking and development takes place (see Sect. 13.3.4). Mathematical thinking and learning cannot be considered as something that happens in the students' isolated mind, independently of external influences. ...
Since the foundation of the Mathematikum, Germany, in 2002 and Il Giardino di Archimede, Florence, Italy, in 2004 there have been many activities around the world to present mathematical experiments in exhibitions and museums. Although these activities are all very successful with respect to their number of visitors, the question arises what is their impact for “learning” mathematics in a broad sense. This question is discussed in the paper. We present a few experiments from the Mathematikum and shall then discuss the questions, as to whether these are experiments and whether they show mathematics. The conclusion will be that experiments provide an optimal first step into mathematics. This means in particular that they do not offer the whole depth of mathematical reasoning, but let the visitors experience real mathematics, insofar as they provide insight by thinking.
... Several papers have summarized the development of research on teachers and teaching in PME in the past (e.g., Hoyles 1992; da Ponte and Chapman 2006; Llinares and Krainer 2006;Jaworski 2011;Lin and Rowland 2016). In most contributions, the development of the field is described by three phases. ...
... In their summary of the field for the 30 years PME volume in 2006, da Ponte and Chapman (2006) identify four main objects of study in teacher-related research on PME: Teachers' mathematical knowledge, Teachers' knowledge of mathematics teaching, Teachers' beliefs and conceptions, Teachers' practices. While all of these topics are still in place, Lin and Rowland (2016) put teacher knowledge in the centre of their contribution for the 40 year handbook ten years later, highlighting its role in PME research. ...
... Advent of so-called "socio-cultural approaches": Several authors offer explanations for the increased focus on the teacher in the late 80s and in the 90s. Lin and Rowland (2016) note that this development coincided with the so-called "social turn", meaning an increased focus on social context, in which students' mathematical thinking and development takes place (see Sect. 13.3.4). Mathematical thinking and learning cannot be considered as something that happens in the students' isolated mind, independently of external influences. ...
This paper explores the design and longitudinal effect of an intervention approach for supporting children who are mathematically vulnerable: the Extending Mathematical Understanding (EMU)—Intervention approach. The progress over three years of Grade 1 children who participated in the intervention was analysed and compared with the progress of peers across four whole number domains. The findings show that participation in the EMU program was associated with increased confidence and accelerated learning that was maintained and extended in subsequent years for most children. Forty per cent of children were no longer vulnerable in the year following the intervention, and others were vulnerable in fewer domains. Comparative data for non-EMU participants highlights the wide distribution of mathematics knowledge across all children in each grade level. This explains why classroom teaching is so complex and highlights the challenges teachers face in providing inclusive learning environments that enable all students to thrive.
... Several papers have summarized the development of research on teachers and teaching in PME in the past (e.g., Hoyles, 1992;da Ponte and Chapman, 2006;Lineares and Krainer, 2006;Jaworski, 2011;Lin and Rowland, 2016). In most contributions, the development of the field is described by three phases. ...
... In their summary of the field for the 30 years PME volume in 2006, da Ponte andChapman, (2006) identify four main objects of study in teacher-related research on PME: Teachers' mathematical knowledge, Teachers' knowledge of mathematics teaching, Teachers' beliefs and conceptions, Teachers' practices. While all of these topics are still in place, Lin and Rowland (2016) put teacher knowledge in the centre of their contribution for the 40 year handbook ten years later, highlighting its role in PME research. ...
... Advent of so-called "socio-cultural approaches": Several authors offer explanations for the increased focus on the teacher in the late 80s and in the 90s. Lin and Rowland (2016) note that this development coincided with the so-called "social turn", meaning an increased focus on social context, in which students' mathematical thinking and development takes place (see 3.4). Mathematical thinking and learning cannot be considered as something that happens in the students' isolated mind, independently of external influences. ...
... Several papers have summarized the development of research on teachers and teaching in PME in the past (e.g., Hoyles 1992; da Ponte and Chapman 2006; Llinares and Krainer 2006;Jaworski 2011;Lin and Rowland 2016). In most contributions, the development of the field is described by three phases. ...
... In their summary of the field for the 30 years PME volume in 2006, da Ponte and Chapman (2006) identify four main objects of study in teacher-related research on PME: Teachers' mathematical knowledge, Teachers' knowledge of mathematics teaching, Teachers' beliefs and conceptions, Teachers' practices. While all of these topics are still in place, Lin and Rowland (2016) put teacher knowledge in the centre of their contribution for the 40 year handbook ten years later, highlighting its role in PME research. ...
... Advent of so-called "socio-cultural approaches": Several authors offer explanations for the increased focus on the teacher in the late 80s and in the 90s. Lin and Rowland (2016) note that this development coincided with the so-called "social turn", meaning an increased focus on social context, in which students' mathematical thinking and development takes place (see Sect. 13.3.4). Mathematical thinking and learning cannot be considered as something that happens in the students' isolated mind, independently of external influences. ...
Since 2004, in accordance with the Federal Educational Standards, probability theory and statistics has been included into teaching practice in Russian schools. This paper focuses on one form of this work: organization of intellectual competitions on probability theory and statistics for school students. Since 2008, the Moscow Center for Continuous Mathematical Education has conducted the Internet Olympiad for students in school years 6–11. In addition to the traditional problems, participants are offered a choice to write an essay on a proposed topic. This article attempts to classify those topics and highlight the most popular ones among the students. In addition, this paper makes a short overview of selected problems that from the organizers’ point of view represent promising and prospective trends in the teaching of probability and statistics at school. The article is addressed to education specialists, teachers, and researchers who specialize in probability theory and statistics.
... En la literatura internacional existe un creciente interés por el diseño de estrategias y ambientes que aporten a la formación de profesores. La formación de profesores a través de la investigación de corte profesional es una de ellas [1,2]. En [2] se afirma que la investigación de problemas específicos de la propia profesión es una poderosa forma de combinar colaboración, práctica y la focalización en el aprendizaje de los estudiantes. ...
... En [2] se afirma que la investigación de problemas específicos de la propia profesión es una poderosa forma de combinar colaboración, práctica y la focalización en el aprendizaje de los estudiantes. En la misma dirección, en [1] anotan que la investigación es uno de los métodos que aporta significativamente al perfeccionamiento de la práctica de los profesores. Para los autores: ...
La formación en investigación es un tema que ocupa las agendas de universidades e instituciones educativas en Colombia y el mundo. En particular, el desarrollo de habilidades, competencias y la formación para el campo profesional en profesores ha sido un objeto de estudio de creciente interés. En esta línea, se presentan resultados de un estudio que buscó identificar las percepciones de futuros profesores de matemáticas frente a las contribuciones de las estrategias implementadas en un Semillero de Investigación a su formación en investigación. Para cumplir con este propósito, se consolidó un grupo focal con integrantes del Semillero, se grabaron sesiones del grupo en audio, que se analizaron siguiendo las orientaciones de un análisis de contenido. Los resultados del estudio se presentan en dos categorías: 1) las oportunidades para la formación en investigación que ofrece un semillero y su relación con otras propuestas, y 2) las actividades y estrategias que ofrece un semillero para la formación en investigación. Frente a esta última se describe un panorama general y se especifican las valoraciones de un recurso metodológico basado en la observación de clase, que se implementó en el Semillero, que los estudiantes estudiaron y se apropiaron críticamente. Los resultados también muestran que los estudiantes valoran positivamente el Semillero como una estrategia de formación diferenciada y complementaria a otros espacios, como la práctica pedagógica; además, reconocen en este espacio reflexiones y discusiones que ayudan a profundizar en contenidos propios de los programas. Los participantes destacan los conversatorios, los talleres y las lecturas compartidas como estrategias que contribuyen a su formación. Finalmente, la estrategia que se empleó permitió que los futuros profesores ampliaran su panorama del aula y reconocieran la necesidad de conocimientos más allá de los disciplinares para desempeñar su labor.
... It constitutes a personal and contextualized space in the classrooms, since what is developed in it, watering in the research in didactics of mathematics to stress the naive and tempting solutions to the complex problems that arise in the classes. At the same time, it is a proactive process because, from the reflection on practice, synergies that allow the incidence of mathematics didactics can be established, to both train teachers and teaching professionals and guide their practice in the classroom (Lin & Rowland, 2016). ...
In this article we share a study on a professional development training program for math teachers. It is presented as innovative as it focuses on the reflection on the practice of teaching through the raising and solution of professional problems. We define professional problems, such as those teaching situations to which the teacher seeks solutions. We demonstrate with excerpts from working sessions that these solutions are nourished by the reflection on the practice they are trying to modify. They are, as a whole, indicators of specialized knowledge that characterize a professional of teaching mathematics.
... Ideally, LS motivates teachers to analyze and reflect on those aspects of the lesson that best help children's mathematical reasoning (Lin & Rowland, 2016), oftentimes guided by seasoned teachers who assume the role of LS facilitators, as it was the case in our sample. In fact, teachers discussed and debated with their LS facilitators a range of decisions at each stage in the LS cycle, from how to group students to what sort of manipulatives to use for the lesson. ...
Viewing teachers as learners of policy reform, this exploratory study examines a group of elementary mathematics teachers as they discussed teaching with multiple strategies as found in the new Mathematics Florida Standards during a lesson study cycle. In particular, it describes how teachers: (a) advance different explanations for teaching with multiple strategies in the new standards, and (b) anticipate or recognize major obstacles to the implementation of these new standards. Considerations of this study’s results to further research on teacher professional development and educational reform are also briefly discussed.
... En la actualidad existen suficientes datos provenientes de estudios realizados en el marco de diversos ámbitos de investigación en educación matemática, como por ejemplo el análisis de la práctica del profesor y el desarrollo profesional, para confirmar que el enfoque de enseñanza repercute en el aprendizaje del alumnado, entre otros muchos aspectos (Charalambous y Pitta-Pantazi, 2016;Lin y Rowland, 2016). ...
El objetivo de este artículo es promover la enseñanza de la estadística en contexto en Educación Infantil y Primaria. Con este propósito, en la primera parte se argumenta la importancia de planificar la enseñanza de la estadística a partir de contextos reales y se ofrecen recursos y estrategias a partir de proyectos y ciclos de investigación estadística, principalmente; en la segunda parte, se describe una investigación estadística en una escuela a partir de la celebración del carnaval. Los resultados muestran una visión longitudinal de los contenidos de estadística desde Educación Infantil hasta 6º de primaria, que se inicia con el recuento de datos a partir de los propios alumnos y finaliza con la representación de datos con diagramas de sectores. Se concluye que es necesario avanzar hacia la definición de una línea metodológica de centro en el área de matemáticas para lograr una enseñanza coherente que promueva la alfabetización estadística en todos los niveles.
... Una muestra del creciente interés en el estudio del conocimiento del profesor es que, de los cuatro focos en los que Ponte y Chapman (2006) organizan su revisión de la investigación presentada en el PME sobre el profesor, dos de ellos se refieren a su conocimiento. Además, en la siguiente revisión de la producción del PME, el capítulo dedicado a la investigación sobre profesores (Lin y Rowland, 2016) se centra en su conocimiento y desarrollo profesional. ...
Resumen: Este artículo, partiendo de la observación de una lección de un profesor chileno de Educación Secundaria en la que se introduce el teorema de Thales, aborda la interpretación de dicha lección desde el conocimiento del profesor, utilizando el modelo Mathematics Teacher's Specialised Knowledge (MTSK). Se consideran aspectos del conocimiento especializado en relación, por un lado, con el teorema de Thales como objeto de aprendizaje y enseñanza, y, por otro, con la práctica matemática de demostrar. Teniendo como referente las propuestas del currículo chileno para la enseñanza de dicho contenido, extraemos una imagen de elementos relacionados del conocimiento del profesor que nos permiten explicar qué se enfatiza en la lección. Los
... El estudio del desarrollo profesional de los profesores de matemáticas ha sido uno de los focos de investigación más activos internacionalmente de los últimos años (Lin y Rowland, 2016). Diversas investigaciones asocian el desarrollo profesional a diferentes factores, como el aumento de coherencia entre las concepciones y la práctica de aula (Kaiser y Li, 2011), la mejora de la capacidad reflexiva (Cobb y McLain, 2001), o el aumento de conocimiento profesional (Bell, Wilson, Higgins y McCoach, 2010). ...
La formación de maestros ya egresados es un área infraexplorada en la investigación en Educación Matemática, especialmente en relación con su conocimiento. Presentamos aquí los resultados de un experimento de enseñanza orientado a la formación de maestros egresados. Este experimento tuvo lugar en la Universidad de Huelva con un total de 39 maestros, en el contexto de un curso de adaptación al Grado de Primaria. Para la fundamentación teórica del experimento usamos el modelo de Conocimiento Especializado del Profesor de Matemáticas, que permitió a su vez generar una hipótesis de progresión de aprendizaje. Mostraremos tanto el diseño del experimento, como el análisis retrospectivo del mismo. Los resultados del estudio evidenciaron que los maestros mejoran en el uso de su conocimiento de los temas, y de la enseñanza de las matemáticas.
... This differentiation reflects claims from the literature that teachers' professional knowledge is, at least to some degree, situation-specific, (e.g. Lin and Rowland, 2016). Diagnosing draws on professional knowledge and aims to accumulate information about students' understanding. ...
Teachers' diagnostic competences are regarded as highly important for classroom assessment and teacher decision making. Prior conceptualizations of diagnostic competences as judgement accuracy have been extended to include a wider understanding of what constitutes a diagnosis; novel models of teachers' diagnostic competences explicitly include the diagnostic process as the core of diagnosing. In this context, domain-general and mathematics-specific research emphasizes the importance of tasks used to elicit student cognition. However, the role of (mathematical) tasks in diagnostic processes has not yet attracted much systematic empirical research interest. In particular, it is currently unclear whether teachers consider diagnostic task potential when selecting tasks for diagnostic interviews and how this relationship is shaped by their professional knowledge. This study focuses on pre-service mathematics teachers' selection of tasks during one-to-one diagnostic interviews in live simulations. Each participant worked on two 30 mins interviews in the role of a teacher, diagnosing a student's mathematical understanding of decimal fractions. The participants' professional knowledge was measured afterward. Trained assistants played simulated students, who portrayed one of four student case profiles, each having different mathematical (mis-) conceptions of decimal fractions. For the interview, participants could select tasks from a set of 45 tasks with different diagnostic task potentials. Two aspects of task selection during the diagnostic processes were analyzed: participants' sensitivity to the diagnostic potential, which was reflected in higher odds for selecting tasks with high potential than tasks with low potential, and the adaptive use of diagnostic task potential, which was reflected in task selection influenced by a task's diagnostic potential in combination with previously collected information about the student's understanding. The results show that participants vary in their sensitivity to diagnostic task potential, but not in their adaptive use. Moreover, participants' content knowledge had a significant effect on their sensitivity. However, the effects of pedagogical content and pedagogical knowledge did not reach significance. The results highlight that pre-service teachers require further support to effectively attend to diagnostic task potential. Simulations were used for assessment purposes in this study, and they appear promising for this purpose because they allow for the creation of authentic yet controlled situations.
... The program had the challenge of meeting the demands of the whole system, all localities, and also of individual teachers. These demands entailed the creation of an instructional model suited to be scaled, which provides a continuous and coherent set of experiences over an extended period of time (Lin and Rowland 2016;Marrongelle et al. 2013), but at the same time allows a certain amount of flexibility according to local needs. ...
Blended learning, which combines face-to-face workshops with self-directed online learning, is becoming a good alternative in designing and deploying professional development programs. The online component adapts to teachers’ time constraints, requires fewer trained professionals for its implementation, and enhances participants’ opportunities to engage in the exploration and visualization of mathematical concepts and ideas. Also, the face-to-face component of this instruction modality allows spontaneous communication and collaborative construction of knowledge in real settings. This paper describes the instructional model of Suma y Sigue, a b-learning professional development program for primary and middle school teachers, which has been designed and implemented in Chile since 2015, aimed at developing mathematical knowledge for teaching. A particular feature of the program is its high-degree of self-directed autonomous online learning. We discuss how contextualized situations are articulated to create a learning environment in which teachers get involved in a progressive construction of multiple components of this knowledge. We also describe the implementation of the program, discussing the effects of scaling in teachers’ experiences, and also how the information gathered upon implementation has led to changes in the training of instructors and in the program’s instructional design.
... Although 36% of articles in the no-disability set focused on teachers, only 9% percent of articles in the disability set did so. Mathematics educational research has spent decades highly focused on the role of the teacher knowledge and development (Lin & Rowland 2016). Teachers are viewed as creative and complex professionals, whose actions impact learning, and are influenced by their knowledge, beliefs, and practice, and its intersections (Charalambous 2015). ...
Using a Disability Studies in Education framework, this systematic review analyzed international research published in English (2013–2017) on the teaching and learning of mathematics from the prekindergarten to 12th-grade level, comparing research on students identified as having disabilities to research on students without disabilities. Coding articles (N = 2477) for methodology, participants, mathematical domain, and theoretical orientation, we found that research on students with disabilities was overwhelmingly quantitative (81%) and tended to use behavioral and medical theoretical orientations. Research on students without disabilities was both qualitative (42%) and quantitative (42%) and tended to use constructivist and sociocultural theoretical orientations. In addition, research on mathematical learning that included students with disabilities lacked sustained qualitative inquiry documenting learning processes of students with disabilities and rarely included the teacher as an explicit focus. Following Gervasoni and Lindenskov (2011), we contend that these pronounced differences in research contribute to the segregation of students with disabilities and low-achieving students in lower quality mathematics instruction and may lead to low expectations of the mathematical competence of students with disabilities. We call for increased attention to research that considers how disability is produced and enacted in the complex context of mathematics classrooms.
... The studies conducted to date focus primarily on teachers as problem solvers, with a paucity of papers addressing PS from the perspective of their knowledge (Lester, 2013). Lin and Rowland (2016) spotlighted a need to enlarge on certain particulars or reinterpret existing teacher knowledge models, deemed to be overly general and to omit PSrelated elements. Chapman (2015) authored one of the more prominent studies on the issue; however, research is still insufficient to determine the utility of the various elements comprising knowledge of PS. ...
This study explored future primary teachers' knowledge of pedagogical orchestration in problem-solving instruction. The tool used was a questionnaire distributed to 77 future teachers who are finalizing their pre-service training, divided into two groups on the grounds of differences in their university training. Our findings showed that the respondents had a suitable theoretical understanding of the approaches to teaching problem solving and the associated practices. In light of the contradictions in some of their replies, however, that understanding may not necessarily be carried over into their classroom delivery.
... These premises can be challenged in multiple ways. Starting with the second premise first, one can point out that ours is a practice-engaged field (e.g., Hiebert, 2013;Lin & Rowland, 2016;Morris & Hiebert, 2015). Hence, an important venue for the dissemination of scholarship is practitioner journals, and the impact of this scholarship is likely to come not in the form of citations but through enactment of the ideas by teachers, instructors of methods courses, teacher leaders, and professional developers. ...
The authors discussed 2 paths that the mathematics education community should consider with regard to citation-based metrics of journal quality: either working within the system to enhance positioning or resisting or modifying the system itself.
... In summary, the findings point out that the PT recognized the value of the investigative work in their learning. This result aligns well with what is also reported for inservice teachers (Lin and Rowland 2016). As Watson and Mason (2007) suggest, we aim to develop a teacher education program consistent with what we consider important for mathematics teaching practice. ...
In teacher education, a key issue is how prospective teachers learn. At the University of Lisbon, based on an inquiry-based approach to mathematics learning, we developed a secondary school mathematics teacher education program, in which a central feature is the elaboration of a final investigative report based on teaching practice. In this paper, our aim is to understand the professional learning opportunities and difficulties recognized by prospective teachers (PT) in their final reports, by addressing the following research questions: (1) What didactical choices do PT mention regarding the approach to teaching, the use of tasks, resources and assessment strategies in their teaching practices? (2) What transversal elements of teaching practices do they recognize as enabling their professional development? and (3) What elements of the investigative work do PT refer to as major professional learning outcomes? Using content analysis, we reviewed, coded, and analyzed all 38 reports produced so far in the program. The results suggest that the PT embraced an inquiry-based approach to learning, valuing the role of suitable tasks and of whole class discussions. They also valued reflection and collaboration as practices that support professional development. Prospective teachers also indicated some difficulties and challenges in doing this investigation based on their teaching practice, but even so they tended to regard it a learning opportunity. We conclude that the final report, by its content, structure, and working processes as framed in this teaching education program, supports participants’ development as teachers who hold research in positive regard
Este libro es un homenaje a los profesores Pablo Flores e Isidoro Segovia, del Departamento de Didáctica de la Matemática de la Universidad de Granada. Su tesón, su compromiso con la calidad y la innovación en la docencia superior y en la investigación en Educación Matemática, y, sobre todo, su empatía y cordialidad con todos sus compañeros durante su etapa de gestión, han sido un ejemplo para todos nosotros. Más aún, su contribución a la organización y la gestión de la Facultad de Ciencias de la Educación ha sido una constante en sus vidas profesionales. Y a todo esto se le añade una sostenida experiencia en centros educativos, en aulas de Educación Primaria y Secundaria, en los centros de profesorado y en la formación de profesores de matemáticas en ejercicio. Todas esas actividades están representadas de una forma u otra en este libro, cuyos capítulos abordan las líneas de trabajo de estos dos profesores. Pablo Flores ha desarrollado su actividad universitaria en torno a la formación de profesores, a los componentes de su conocimiento y desarrollo profesional y a sus sistemas de creencias. La de Isidoro Segovia se ha centrado en los procesos de estimación y cálculo, la resolución e invención de problemas, la noción de currículo y la formación de profesores. En este libro participan 44 autores de 15 instituciones educativas de España, Chile, Colombia y Costa Rica. El objetivo de esta obra es dedicarles un merecido reconocimiento por su trabajo desde el cariño, la admiración y el respeto más profundos.
Desde el siglo XIX la educación ha sido la base para que las personas creen una vida para ellas y sus familias, y se conviertan en ciudadanos activamente comprometidos con el Planeta. En la Nueva Era se da por sentado que los niños comienzan la escuela alrededor de los cinco años y pasan por no menos de 11 años de escolaridad obligatoria. Sin embargo, si bien el objetivo de la educación es preparar a los estudiantes para tener éxito en la vida como profesionales, y aunque el mundo en este siglo está pasando por cambios inimaginables hace apenas dos décadas, el sistema educativo todavía no se ha adaptado y la escuela sigue aplicando metodologías que funcionaban cuando los trabajos rutinarios tenían amplia demanda. Por eso, el compromiso de los autores en este libro es por un sistema de educación generalizado, que se innove y actualice de la mano con el crecimiento y la prosperidad del conocimiento y del desarrollo de la humanidad.
The Mathematical Working Space (MWS) theory will be discussed in relation to the characterisation of the professional development of teachersTeacher trainingProfessional development of teachers, and how it can be made operational. The cross-case comparative analysis reported in this chapter is built on situational elements (different countries and context) and models about professional development of teachersTeacher trainingProfessional development of teachers or those about teacher’s mathematical knowledgeMathematical knowledge for teaching found in the specific case analysis reports since 2012 in several MWS Symposia or journals. Considering this plurality of the theoretical model using and comparing it to MWS, we take advantage of the notion of the development of mathematical work to make sense of interplay between epistemological and cognitive plans and variables related to the specialised knowledge of mathematics and teaching.
Developing teachers’ professionalism is important to developing the internal capacity of schools in rural and remote areas. This study investigated rural mathematics educators’ (n = 677) perceptions of professionalism to better understand how to provide more focused and relevant support for mathematics teachers in similar rural and remote contexts. A survey was administered to workshop participants and electronically to teachers across the selected state who did not participate in the workshop. The findings indicate that there are some differences in perception related to continuing to develop as professionals for teachers who choose to participate in professional development workshops; they tended to be more aware and supportive of considering pedagogical changes and learning more mathematical content. Additionally, rural teachers tended to view instructional coaching as a luxury and to consider improvements in pedagogical and content knowledge to be not as essential. Implications for practice, including rural practice, are also addressed.
This study explores the knowledge required for teachers to teach problem solving (PS) from a Primary Mathematics Curriculum Guidelines perspective. It analyzes six countries’ curricular guidelines for primary education using the Mathematical Problem-Solving Knowledge for Teaching model. To identify the PS knowledge required in each education system, the country guidelines were selected based on the country’s results in the 2012 Programme for International Student Assessment (PISA) survey. Data analysis revealed that PS-related knowledge included in the curricula is broad and challenging for teachers. Further, it is not always coherent and research-based. More specifically, the findings show that curricular guidelines emphasize problem classification and solving processes. Our analysis supports the conclusion that particularities in teachers’ knowledge become visible when we view it from the perspective of PS rather than of mathematical concepts.
La literatura reporta la necesidad de programas que aporten a la formación de los profesores en Educación STEM. Debido a tensiones como la caracterización de una disciplina en el contexto de las demás y el delimitar los aprendizajes específicos de cada disciplina en una implementación interdisciplinar, existe un especial interés en los profesores de matemáticas y ciencias pues enfrentan más tensiones al favorecer la educación STEM por los fundamentos epistemológicos de dichas disciplinas. En el estudio que se reporta en este documento se ofrece una aproximación al componente teórico de un programa de formación de profesores de matemáticas y ciencias en educación STEM. A partir de una revisión de literatura, se identificaron características fundamentales y nucleares en el componente teórico del programa: los conocimientos a desarrollar, las experiencias a beneficiar y el favorecimiento de una configuración de comunidades de aprendizaje profesional.
La formación de los profesores puede considerarse como un aspecto clave en el desarrollo y calidad de los procesos educativos; también como un área/campo de investigación en el que confluyen los desarrollos en la investigación educativa y en las disciplinas académicas. En las últimas dos décadas, la investigación sobre los aprendizajes de los profesores de matemática ha tenido significativos desarrollos que incluyen la delimitación de constructos, enfoques y programas para el desarrollo profesional, modelos para describir el conocimiento del profesor, sus creencias, competencias y más recientemente, estudios que se centran en programas para la formación de profesores y en la formación del formador de profesores. Este documento ofrece una introducción (Editorial) del segundo número del volumen 19 de la revista Uni-pluriversidad que se dedicó a la investigación en formación profesional de profesores de matemáticas.
Professional knowledge is highlighted as an important prerequisite of both medical doctors and teachers. Based on recent conceptions of professional knowledge in these fields, knowledge can be differentiated within several aspects. However, these knowledge aspects are currently conceptualized differently across different domains and projects. Thus, this paper describes recent frameworks for professional knowledge in medical and educational sciences, which are then integrated into an interdisciplinary two-dimensional model of professional knowledge that can help to align terminology in both domains and compare research results. The models’ two dimensions differentiate between cognitive types of knowledge and content-related knowledge facets and introduces a terminology for all emerging knowledge aspects. The models’ applicability for medical and educational sciences is demonstrated in the context of diagnosis by describing prototypical diagnostic settings for medical doctors as well as for teachers, which illustrate how the framework can be applied and operationalized in these areas. Subsequently, the role of the different knowledge aspects for acting and the possibility of transfer between different content areas are discussed. In conclusion, a possible extension of the model along a “third dimension” that focuses on the effects of growing expertise on professional knowledge over time is proposed and issues for further research are outlined.
The goal of this study aims at introducing an entry course of a 3-year sequential courses module for a secondary mathematics teacher education program in Taiwan. This module is a reformed teacher education curriculum planned for Prospective Secondary Mathematics Teachers (PSMTs) to learn how to teach with the field-study approach. The field-study approach provides abundant opportunities for PSMTs to cultivate their competencies in teaching. In this chapter, we take the first year course to deliberate why the Psychology of Mathematics Learning is selected as an entry course for the teacher education program and how it works. Considering the importance to raise PSMTs’ awareness of students’ mathematical thinking and to cultivate their competencies of sensitizing students’ mathematical thinking, and ultimately to bear the competencies as the habitus in their future teaching professional, the mission of the course focuses on PSMTs’ learning of understanding students’ mathematical thinking through the process of cyclic learning. The quality of PSMTs dynamic learning in the field study can be evaluated by their study work. This chapter provides one example of PSMTs’ survey study in one complete learning cycle, and summarizes several criteria of evaluating how PSMTs conduct a study to understand students’ mathematical thinking in a holistic perspective.
In previous work, we have outlined points that appeared significant for the
mathematical education of mathematics teachers. Issues of school mathematics, in
opposition to academic mathematics, and of engaging teachers in a practice of
mathematizing (Bauersfeld, 1998), in opposition to being exposed to standardized
knowledge, were argued as fundamental dimension of an approach better articulated
with mathematics teaching practices. A two-year project was developed along that
perspective with a group of secondary mathematics teachers. To illustrate the
potential of the approach, we analyse the resources used and activated by teachers
when making sense of a mathematical situation proposed by the teacher educators.
This paper presents a professional development project, conducted with teachers, focusing on the exploration of the mathematics related to their teaching practice. We clarify the approach developed and present findings about the mathematical comprehensions emerging in situ. One main outcome is that in addition to restructuring their understandings of the mathematical concepts they teach everyday, teachers engaged in deep mathematical reflections grounded in their teaching, outlining repercussions of these explorations for their practice.
Teachers try to help their students learn. But why do they make the particular teaching choices they do? What resources do they draw upon? What accounts for the success or failure of their efforts? In How We Think, esteemed scholar and mathematician, Alan H. Schoenfeld, proposes a groundbreaking theory and model for how we think and act in the classroom and beyond. Based on thirty years of research on problem solving and teaching, Schoenfeld provides compelling evidence for a concrete approach that describes how teachers, and individuals more generally, navigate their way through in-the-moment decision-making in well-practiced domains. Applying his theoretical model to detailed representations and analyses of teachers at work as well as of professionals outside education, Schoenfeld argues that understanding and recognizing the goal-oriented patterns of our day to day decisions can help identify what makes effective or ineffective behavior in the classroom and beyond.
The following is intended as an analysis of a specific conceptual change in mathematics during the early 19th century. From this some pedagogical conclusions are drawn. Analyzing a special example seems to be a more appropriate way of handling the relevance of historical and philosophical considerations for mathematical education than dealing with this question in general terms. My example refers to the fact that at the turn of the 19th century one began to distinguish more systematically between numbers and quantities than had been done before. The program of arithmetizing mathematics arose. Certainly this process had various causes. In the following I concentrate on one of these, namely, the relations of mathematics to experimental sciences.
This Research Forum highlights the most recent research on the development of the role of the teacher of mathematics within mathematics classrooms that involve the use of technological tools, with an emphasis on teachers’ experiences within both formal and informal professional development programmes. We foreground the theoretical ideas and methodological approaches that focus on the development of classroom practices at the levels of both individual teachers and communities of teachers, charting their respective development over time. The RF makes reference to a previous forum at PME37 on the theme of Meta-Didactical Transposition (Aldon et al. 2013a), a theoretical framework that has evolved from research in this area.
The aim of this paper is to a) shed light on the nature of student teachers' noticing of mathematics specific phenomena as observed in a video recorded lesson and to b) compare this nature for student teachers at the beginning of their master studies at the university and those at its end. Our study is based on a thorough examination of student teachers' written analyses (n = 169) of video recorded lessons. We capture the qualities of these in terms of the author-defined notion of mathematics specific (or MS) phenomena by a) matching the students' comments against what we view as important issues in the lessons, and b) developing a framework to further characterise the nature of the observations. Both qualitative and quantitative results corroborate the findings of earlier research on pre-service teachers' lesson analyses in that they pay limited attention to content in the lesson observed. Moreover, it transpires that students tend to notice MS phenomena which are not identified as important by experts and that the demonstrated ability to notice MS phenomena does not show significant differences for students in two distinct stages of a teacher preparation programme.
This Research Forum highlights the most recent research on the development of the role of the teacher of mathematics within mathematics classrooms that involve the use of technological tools, with an emphasis on teachers’ experiences within both formal and informal professional development programmes. We foreground the theoretical ideas and methodological approaches that focus on the development of classroom practices at the levels of both individual teachers and communities of teachers, charting their respective development over time. The RF makes reference to a previous forum at PME37 on the theme of Meta-Didactical Transposition (Aldon et al. 2013a), a theoretical framework that has evolved from research in this area.
Task-related research formats may afford highly practice-relevant insight into teachers' professional knowledge and views. This study explores what kinds of tasks in-service teachers think of when aiming to assess their students' conceptual understanding of fractions. In a top-down coding approach taking into account a sample of 87 teachers a particular focus was put on core aspects of fractions addressed by the teachers and on requirements regarding conversions of representations. Moreover, possible interrelations of such content domain-specific PCK with the teachers' pedagogical content views on teaching and learning mathematics were considered. The results indicate that most teachers focused only on a few core aspects of fractions and they suggest interrelations with more global views.
This paper presents a case study of one Year 6 teacher who, over a period of one year, developed a deeper understanding of equivalent fractions. Evidence from assessment of teacher pedagogical content knowledge, interviews and classroom observation reveals that the exploration of representations of fractions was central to the teacher's growth in knowledge for teaching. Also apparent was a change in pedagogy, from a focus on procedural understanding to an emphasis on developing conceptual understanding. We propose that the deliberate exploration of representations may be a key to increasing both the mathematical and pedagogical knowledge of teachers, supporting their ability to foster conceptual understanding in their students.
The aim of this paper is to a) shed light on the nature of student
teachers’ noticing of mathematics specific phenomena as observed in a video recorded lesson and to b) compare this nature for student teachers at the beginning of their master studies at the university and those at its end. Our study is based on a thorough examination of student teachers’ written analyses (n = 169) of video recorded lessons. We capture the qualities of these in terms of the author-defined notion of mathematics specific (or MS) phenomena by a) matching the students’ comments against what we view as important issues in the lessons, and b) developing a framework to further characterise the nature of the observations. Both qualitative and quantitative results corroborate the findings of earlier research on pre-service teachers’ lesson analyses in that they pay limited attention to content in the lesson observed. Moreover, it transpires that students tend to notice MS phenomena which are not identified as important by experts and that the demonstrated ability to notice MS phenomena does not show significant differences for students in two distinct stages of a teacher preparation programme.
This paper sets forth a way of interpreting mathematics classrooms that aims to account for how students develop mathematical beliefs and values and, consequently, how they become intellectually autonomous in mathematics. To do so, we advance the notion of sociomathematical norms, that is, normative aspects of mathematical discussions that are specific to students' mathematical activity. The explication of sociomathematical norms extends our previous work on general classroom social norms that sustain inquiry-based discussion and argumentation. Episodes from a second-grade classroom where mathematics instruction generally followed an inquiry tradition are used to clarify the processes by which sociomathematical norms are interactively constituted and to illustrate how these norms regulate mathematical argumentation and influence learning opportunities for both the students and the teacher. In doing so, we both clarify how students develop a mathematical disposition and account for students' development of increasing intellectual autonomy in mathematics. In the process, the teacher's role as a representative of the mathematical community is elaborated.
In this article we report on an exploratory study on development of pre-service teachers' competences in using different representations of multiplication. Grounded in the literature on pedagogical and psychological value of knowing and using representations we investigate student teachers preferences of representations of multiplication in the case where they are given free choice. A survey was distributed to 121 prospective elementary school teachers at University. The analyses of their answers highlighted differentiated choice of representations made by students. The choice of preferred representation was linked to problem abstractness.
Professional knowledge of mathematics teachers related to big ideas in mathematics and mathematics instruction can enhance teachers' competencies of designing rich learning opportunities in the mathematics classroom. Responding to a need of empirical research on professional knowledge connected with big ideas in mathematics, this study presents results of a test administered to more than 100 German pre-service teachers. The results indicate that the pre-service teachers often were unable to discern big ideas behind mathematical contents and to link elements of content matter according to these big ideas. The results call for an emphasis in teacher education not only on a solid content matter knowledge base, but also on overarching concepts and meta-mathematical ideas.
This study investigates Chinese and U.S. teachers' constructions of pedagogical representations by analyzing the video-taped lessons from the Learner's Perspective Study, involving 10 Chinese and 10 U.S. consecutive lessons on the topic of linear equations or linear relations. This study allows not only for the examination of what pedagogical representations Chinese and U.S. teachers construct, but also for the examination of the changes and progressions of constructed representations in these Chinese and U.S. lessons. This study is significant because it contributes to our understanding of the cultural differences involving U.S. and Chinese students' mathematical thinking and has practical implications for constructing pedagogical representations to maximize students' learning.
In a quasi-experimental study with 619 students from 29 classrooms (grades 7/8) we investigated the effects of a teacher training on teachers' mistake-handling activities and students' learning of reasoning and proof in geometry. Teachers of the experi-mental group classrooms received a combined training in mistake-handling and teaching reasoning and proof, whereas the teachers of the control group classrooms only took part in a training on teaching reasoning and proof. Their students participated in a pre-and post-test. Moreover, they were asked to evaluate how the teachers handled their mistakes. Our findings show that the teacher training was successful: the teachers of the experimental group classrooms changed their mistake-handling behavior and, compared to the control group classrooms the students in the experimental group performed significantly better in the post-test.
Although the importance of mathematical problem solving is now widely recognised,
relatively little attention has been given to the conceptualisation of mathematical
processes such as representing, analysing, interpreting and communicating. The
construct of Mathematical Knowledge for Teaching (Hill, Ball & Schilling, 2008) is
generally interpreted in terms of mathematical content, and in this paper we describe
our initial attempts to broaden MKT to include mathematical process knowledge
(MPK) and pedagogical process knowledge (PPK). We draw on data from a
problem-solving-focused lesson-study project to highlight and exemplify aspects of the
teachers’ PPK and the implications of this for our developing conceptualisation of the
mathematical knowledge needed for teaching problem solving.
Dealing with representations and changing between them plays a key role for both mathematics as a discipline and for building up mathematical knowledge in the classroom. Hence, professional knowledge and views of teachers related to using multiple representations can be considered as a prerequisite for creating conceptually rich learning opportunities. However, specific empirical research is scarce – in particular there is a lack of studies taking into account that culture might influence such views. Consequently, this study focuses on views about using multiple representations held by more than 100 British and more than 200 German pre-service teachers. The results indicate that culture might influence the views of the pre-service teachers, but also that there are common needs for further professional development.
This paper analyses data from a PhD pilot study to explore the nature of mathematical knowledge for teaching using technology, as represented by the central construct of the TPACK framework. The case study of teacher Alice is used as an illustrative example to suggest that the central TPACK construct may be better understood as a transformation and deepening of existing mathematical knowledge rather than as a new category of knowledge representing the integration of technology, pedagogical and mathematical knowledge.
The aims of DG12 were to: Facilitate discussion of key issues related to the knowledge required by mathematics teacher educators (MTEs), Identify different emergent strands in research that can be related to this area.
Lesson study is a popular professional development approach in Japan whereby teachers collaborate to study content, instruction, and how students solve problems and reach for understanding in order to improve elementary mathematics instruction and learning in the classroom. This book is the first comprehensive look at the system and process of lesson study in Japan. It describes in detail the process of how teachers conducted lesson study--how they collaborated in order to develop a lesson, what they talked about during the process, and what they looked at in order to understand deeply how students were learning. Readers see the planning of a mathematics lesson, as well as how much content knowledge the teachers have. They observe students' problem solving strategies and learn how Japanese teachers prepare themselves to identify those strategies and facilitate the students' discussion. Written for mathematics teachers, educational researchers, school administrators interested in teachers' professional development, and professional developers, this landmark volume provides an in-depth understanding of lesson study that can lead to positive changes in teachers' professional development and in teaching and learning in the United States. © 2004 by Lawrence Erlbaum Associates, Inc. All rights reserved.
This paper states three claims dealing with the relationship between mathematics education researchers and mathematics teachers: (1) Mathematics education research is a highly diverse field; (2) Teachers have various roles as stakeholders in mathematics education research; (3) Regarding teachers as stakeholders in mathematics education research affords reflecting some (fruitful) "cultural differences". The paper claims the necessity to regard researchers as key stakeholders in practice, and teachers as key stakeholders in research. © 2014, The Author(s) & Dept. of Mathematical Sciences-The University of Montana.
In this paper and in accordance to the goals of this book, we provide information on relevant research on mathematics teachers’ learning produced by the PME community. We summarise trends/key issues, contrast or compare perspectives, analyze results, and suggest directions for future research.
We regard mathematics teachers’ learning as a lifelong learning process which starts with one’s own experiences of mathematics teaching from the perspective of a student, or even with mathematical activities before schooling. Research assumes that theses early experiences have a deep and longlasting impact on teachers’ careers. “Teachers tend to teach in the way they have been taught”, is an often used statement. However, pre- and in-service teacher education are important interventions in supporting teachers’ growth. The teachers play the key role in that learning process. They are regarded as active constructors of their knowledge and thus encouraged to reflect on their practice and to change it where it is appropriate. It is a challenge to find answers to the questions of where, how and why teachers learn, taking into account that teachers’ learning is a complex process and is to a large extent influenced by personal, social, organisational, cultural, and political factors.
In our paper we regard three types of teachers: student teachers, teachers, and also teacher educators. We use this distinction as a way of structuring our paper. In section 2 we thus focus on student teachers’ learning and regard PME papers that investigate student teachers’ growth in beliefs and knowledge, including the impact of first experiences in teaching. In section 3 we report about PME papers and activities that put an emphasis on inquiry into practising teachers’ growth when participating in teacher education programmes, underlining the increase of social dimension in framing our understanding of teacher learning. We sift out major goals of these programmes as well as key factors that promote or hinder teachers’ learning. Section 4 refers to a domain which needs closer attention in the future, namely our own learning as teacher educators. It is the field where theory and practice of teacher education inevitably melt together and we thus face the challenge of self-applying our demands on teacher education. Finally, in section 5, we aim at working on key issues that we found in analysing PME research on mathematics teacher education, and we suggest directions for future research in teachers’ learning.
The Technological Pedagogical Content Knowledge (TPACK) framework is increasingly in use by educational technology researchers. The framework provides a generic description of the knowledge requirements for teachers using technology in all subjects. This paper describes the development of a mathematics specific version of the TPACK framework. We show how a particular conception of the knowledge required to teach mathematics can be integrated with the TPACK framework as the basis for understanding technology integrated mathematics teaching. The resulting framework provides a sharper lens than the generic TPACK framework alone and a better understanding of the knowledge required to use technology in teaching mathematics
This chapter examines contemporary frameworks for analysing teacher expertise which are relevant to the integration of digital technologies into everyday teaching practice. It outlines three such frameworks, offering a critical appreciation of each, and then explores some commonalities, complementarities and contrasts between them: the Technological, Pedagogical and Content Knowledge (TPACK) framework (Koehler & Mishra, Contemporary Issues in Technology and Teacher Education, 9(1), 2009); the Instrumental Orchestration framework (Trouche, L. (2005). Instrumental genesis, individual and social aspects. In D. Guin, K. Ruthven, & L. Trouche (Eds.), The didactical challenge of symbolic calculators: Turning a computational device into a mathematical instrument (pp. 197–230). New York: Springer.); and the Structuring Features of Classroom Practice framework (Ruthven, Education & Didactique, 3(1), 2009). To concretise the discussion, the use of digital technologies for algebraic graphing, a now well established form of technology use in secondary school mathematics, serves as an exemplary reference situation: each of the frameworks is illustrated through its application in a study of teacher expertise relating to this topic (respectively Richardson, Contemporary Issues in Technology and Teacher Education, 9(2), 2009; Drijvers, Doorman, Boon, Reed, & Gravemeijer, Educational Studies in Mathematics, 75(2), 213–234, 2010; Ruthven, Deaney, & Hennessy, Educational Studies in Mathematics, 71(3), 279–297, 2009).
This chapter consists of two parts. First, based on a longitudinal experiment aiming at effective mathematics teaching and learning in China, a theory, called teaching with variation, is summarized by adopting two concepts of variation, i.e. conceptual variation and procedural variation. Secondly, it is demonstrated that the Chinese theory is strongly supported by several well-known Western theories of learning and teaching. Particularly, the Marton's theory offers an epistemological foundation and conceptual support for the Chinese theory. Moreover, the authors argue that the teaching with variation characterizes the mathematics teaching in China and by adopting teaching with variation, even with large classes, students still can actively involve themselves in the process of learning and achieve excellent results.
In this paper I present findings from a completed four-year developmental research project in which the Knowledge Quartet (KQ) framework was used as a tool to identify and to support development of the mathematical content knowledge of a group of early career elementary teachers. I focus on the propositional knowledge (Shulman, 1986), of one participant as revealed through the Foundation dimension of the KQ during observations and discussions of her teaching. These findings suggest that supported reflection on teaching which is focused on mathematical content may enhance the development of mathematical content knowledge for teaching.
Our research in the last decade has been into classroom situations that we perceive to make demands on mathematics teachers' disciplinary knowledge of content and pedagogy. Amongst the most visible of such situations are those that we describe as contingent, in which a teacher is challenged to deviate from their planned agenda for the lesson. Our research has been grounded in classroom practice, and our findings and the associated grounded theories have been open to enhancement and revision in the face of new classroom data. We propose a classification of the origins of contingent classroom episodes: namely the students; the teacher him/herself; and pedagogical tools and resources.
This paper aims to explore the educative power of an experienced mathematics teacher educator-researcher (MTE-R) who displayed his insights and strategies in teacher professional development (TPD) programs. To this end, we propose a framework by first conceptualizing educative power based on three constructs—communication, reasoning, and connection—and then we extend the conceptualization with another two dimensions: the reciprocal facilitator-learner relationships involving educators, teachers, and students, as well as a bridge between research and practice. Based on both self-study and case-study approaches, we further elaborate features specific to the MTE-R’s educative power which includes communication using an approach of creating educative phenomenology, reasoning by mapping teachers’ ideas onto emergent models to solve problems in educative challenges, and connection between research and practice by coordination. In particular, the core of the educative power that supported the MTE-R to initiate at-the-moment actions was his insights into the essence of mathematics, and the learning of students and teachers. We believe that the conceptual framework in this study offers a powerful tool that could guide the analyses of educative power, especially for those studies related to the initiation of at-the-moment actions and the implementation of TPD programs.
In this paper, we document some developments in teacher education practice at one university, brought about by reflection on research into mathematics teacher knowledge. The authors are three members of the Cambridge-based research team who developed the Knowledge Quartet (KQ), a theory of mathematics teacher knowledge, with a focus on classroom situations in which this knowledge is applied. At the same time as being researchers, the authors were elementary mathematics teacher education instructors. They found that the KQ research brought about new awareness of the importance of some components of mathematics didactics, as well as providing new tools for undertaking some aspects of their teacher educator role. The paper explores some of these awarenesses and tools in detail.
This paper discusses the nature and role of preservice secondary mathematics teachers' knowledge that supports their use of inquiry approaches during their practicum teaching. It highlights how four categories of teachers' knowledge and, more importantly, the connectedness among them based on a common theme influenced the preservice teachers' use of inquiry approaches and their ability to transform pedagogical theory to practice. The paper also addresses the importance of learning experiences in teacher education that treat these domains of teacher knowledge in an integrated way within classroom-based contextual situations in order to facilitate the development of an appropriate, usable network of knowledge. This study investigated the nature and role of preservice secondary mathematics teachers' knowledge that supported their use of inquiry approaches during their practicum teaching. It is part of an ongoing four-year longitudinal study of beginning secondary mathematics teachers' growth.
INTRODUCTION Superficially, the four articles of this Research Forum may appear widely divergent. However, in this discussion I articulate deeper undercurrents that can contribute to closing an enduring, disturbing, and mostly unspoken gap—pointed out in Simon's article—between what teachers actually and could potentially learn through teaching. Indeed, these articles stress that teachers' craft (planning, implementing, interacting, reflecting, assessing, etc.) can serve as a strategic site for their learning. Concerning this assumed but frequently unrealized potential, I make a twofold argument. The first part of my argument is that articulating both what and how teachers can learn through teaching (LTT) is dearly needed for better understanding this gap and bringing the potential to fruition. This is consistent with Borba's concluding comment of the need to articulate what makes certain experiences of mathematics teachers conducive to the substantial learning they must pursue, so that their students acquire the demanding, reform-oriented expectations of "understanding/doing math." The second part of my argument is that theoretical accounts of teacher learning are required to determine what makes particular learning opportunities productive. I point out how the recently elaborated framework of learning a new mathematical conception through reflection on activity-effect relationship (Ref*AER) (Simon, Tzur, Heinz, & Kinzel, 2004), provides a good basis for such a theoretical account, though adaptations to the complexities of teacher learning will most likely be needed. To substantiate this argument, I briefly present key constructs of the Ref*AER account. Then, I analyze and synthesize elements of the four articles in keeping with the two leading questions: What might teachers LTT that is worthwhile learning (i.e., likely to benefit student learning)? and How might teachers learn this? Finally, I point to ample key issues the four articles raise that await further discussion.
In this paper, we mainly investigate, through the teaching Critical Incident of Practice (CIP), the ways mentors intervene in the mathematics teaching of practice teachers, and the principles and underlying values for their interventions, based on case studies of a group of 8 mentor-practice teachers and their students in secondary schools from the first-year data of a 3-year longitudinal study. The preliminary results show that the principles and ways of mentors' interventions were varied, and they developed frameworks of decision-making in mentoring closely related to the specific modes of intervention that they chose. We expect that both mentors and practice teachers are learning-to-see in mentoring, and developing their professional powers through the co-learning cycle of teaching and mentoring.
Referring to the significant factors affecting teacher quality, "teacher efficacy" deserves to be in the heart of this dilemmatic evolution. The purpose of this study aimed to examine beginning teachers' sense of efficacy in elementary schools, as well as its influential factors. Beginning teachers whose background were and were not in mathematics and science were compared to explore the differences of their teacher efficacy. According to research findings, we should devote all efforts to establish a positive and effective learning organization in order to promote their teacher efficacy internally, externally, and promptly starting from the beginning year.
The study was to develop a mentoring program and examine its effect on mentoring mathematics teaching. A collaborative mentor study group consisting of four mentors and the researcher was set up. The course with 78 hours to develop mentors' theoretical and professional knowledge in which underpins mentoring practice was carried out in the half-year internship. Two surveys, pre-and post-test of pedagogy, self-assessment in mentoring, interview, classroom observation, and reflective journal were the data collected for the study. The satisfaction with the initiates, improvement of mentoring knowledge, and the transfer from the program to support interns on questioning, problem-posing, and anticipating students' solutions were as a result of the mentoring program.
In addressing the need for teachers to be equipped with the kinds of knowledge of mathematics needed to teach in the vision of current mathematics education reforms occurring in the United States, it is important that teachers have support to learn about and improve them. Curriculum materials seem well situated to provide ongoing learning opportunities for teachers because curriculum materials are ubiquitous in classrooms. This paper presents an exploration of the role of teaching experience with reform-oriented, mathematically-supportive curriculum materials (that have the potential to be educative for teachers as well as students) in supporting teachers' learning of mathematics.
This synthesis is designed to provide insight into the most important issues involved in a large-scale implementation of inquiry-based learning (IBL). We will first turn to IBL itself by reflecting on (1) the definition of IBL and (2) examining the current state of the art of its implementation. Afterwards, we will move on to the implementation of IBL and look at its dissemination through resources, professional development, and the involvement of the context. Based on these theoretical reflections, we will develop a conceptual framework for the analysis of dissemination activities before briefly analyzing four exemplary projects. The aim of our analysis is to reflect on the various implementation strategies and raise awareness of the different ways of using and combining them. This synthesis will end with considerations about the framework and conclusions regarding needed future actions.