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Exploring experiences for assisting primary pre-service teachers to extend their knowledge of student strategies and reasoning
Abstract
Exploring how students learn mathematics and what types of tasks promote mathematical reasoning may assist pre-service teachers to develop their pedagogical content knowledge (PCK) for teaching. This study reports on an experience of a cohort of Australian primary (elementary) pre-service teachers, destined to become specialist mathematics teachers associated with the topics of measurement and geometry. The first author presented a lesson with Year 5/6 students related to geometric reasoning and calculating the size of angles. The lesson was launched with no instructions and students were expected to attempt a problem without using a protractor or help from the teacher. The pre-service teachers, classroom teacher and second author were non-participant observers. Findings suggest course experiences provided an opportunity to extend pre-service teachers' knowledge of how students learn and their knowledge of lesson structure and PCK whilst also considering how students might learn to reason and solve a challenging mathematical task.
... Several studies suggest that in the preservice education of teachers who teach mathematics, tasks involving geometric thinking are developed so that prospective teachers can reflect and project the work with this theme in their future professional practice (Brunheira & Ponte, 2019;Erdogan, 2020;Livy & Downton, 2018). Geometry is a system of representation used to visualise concepts, forms of reasoning, and spatial environments (Battista, 2007). ...
... Teachers report that during their education, geometry was reduced to recognising geometric figures, using meaningless formulas and procedures, and working with metric geometry without, for example, distinguishing figural aspects from geometric concepts, finally, without having experienced a geometry teaching that allowed them to think geometrically (Nacarato & Passos, 2003). Livy and Downton (2018) argue that mathematics teachers' preservice education should include situations in which the prospective teachers not only develop their geometric thinking but also discuss pedagogical approaches that support the development of their students' geometric thinking. In this sense, some researchers (Brunheira & Ponte, 2019;Ferreira & Barbosa, 2013) highlight the importance of creating formative spaces capable of promoting interactions between the educator and the prospective mathematics teachers (PMTs), so that they can verbalise their reasoning, debate divergent ideas, build arguments, in other words, actively engage in the construction of geometric knowledge. ...
... Considering that geometry promotes logical thinking and mathematical understanding, mathematics teachers play a crucial role in the teaching and learning process of this theme (van Hiele, 1999). Therefore, knowledge from a theoretical perspective, to develop geometric thinking -such as van Hiele's, for example -in preservice education, can not only expand PMTs' geometric thinking but also the search for ways to support this type of thinking of their future students (Livy & Downton, 2018;Nacarato & Passos, 2003). ...
Background: The study of geometric thinking in the preservice education of mathematics teachers is an emerging theme that can reverberate in the teaching of geometry in basic education. Objectives: To analyse reflections manifested by prospective mathematics teachers (PMTs), working with tasks supported by van Hiele theoretical model to develop geometric thinking. Design: The nature of this study is qualitative and interpretative. Setting and participants: Twenty-four PMTs members of a geometry teaching subject were investigated in a mathematics degree course at a public university in Paraná-Brazil. Data collection and analysis: The data was collected from the video-recorded training sessions, the written production of the PMTs promoted by the tasks and the registers kept on the field diary. The analysis focused on the reflections expressed by PMTs regarding the work with tasks involving geometric thinking, considering the levels of reflection proposed by Muir and Beswick (2007). Results: The results show descriptive, deliberate, and critical reflections, with different levels of incidence, associated with (I) the levels of thought proposed in the van Hiele model; (II) the teacher's role in classroom practice; and (III) the geometric concepts and properties of flat figures. Conclusions: The promotion of formative actions that privilege discussions and reflections on geometric thinking can allow PMTs to seek connections between knowledge of geometry, geometric thinking, and their future teaching practice.
... Recognition of the practical nature of teacher knowledge (Ball et al., 2008) and the complexity of the knowledge required from them (Livy & Downton, 2018) have triggered numerous studies on initial training (Ponte & Chapman, 2006). Those studies seek, mainly, to understand what knowledge prospective teachers should develop to teach -which is an aspect that should not be considered on the sidelines of practice (Ponte & Chapman, 2006). ...
... Given the MR importance as an essential process to support students' mathematics learning, some of those studies focus on prospective teachers' knowledge to identify students' MR and promote their development (e.g., Livy & Downton, 2018;Maher et al., 2014;Stylianides et al., 2013). In addition, MR is a somewhat complex topic, both in terms of its meaning and the variety of associated processes (Herbert et al., 2015). ...
... Teachers' understanding of MR is diverse and sometimes limited -it ranges from merely corresponding to 'thinking' the idea of making, justifying, and validating conjectures and establishing connections between different mathematical ideas (Herbert et al., 2015). However, knowledge and understanding of MR are fundamental to promoting its development in students (Herbert et al., 2015;Livy & Downton, 2018), having implications, for example, on what teachers can accept as a valid justification and how they can support students' reasoning (Livy & Downton, 2018). ...
Background: Mathematical reasoning is fundamental for mathematics learning from the first years of schooling. It is a challenge for students and teachers, so it is relevant to deepen ways to develop this ability with the prospective teachers. Objectives: Identify the challenges in supervised practice with a view to developing students' mathematical reasoning, seeking to answer the following question: What challenges do prospective teachers face in planning and exploring tasks that promote mathematical reasoning? Design: It is based on a formative experiment and follows an interpretive methodology. Setting and participants: This experiment was conducted during 13 sessions of the curricular unit (CU) Didactics of Mathematics of the 2nd year of the master's course in PreSchool Education and Teaching of the 1st Cycle of Basic Education, in a class of 25 students. The participants were four students-two pairs in teaching practice-whose selection followed the following criteria: not having any of the researchers as teaching practice supervisors and regularly intervening in class. Data collection and analysis: the data were collected through participant observation of the CU classes, interviews, and document collection. Results: Students face more challenges associated with mathematical reasoning during monitoring phases of the exploration of the tasks and their final discussion. Conclusions: These results point to the need for initial training programs to prioritise activities that support prospective teachers in the understanding of mathematical reasoning processes and that involve them in the planning of tasks and analysing practical exploration that will enhance their development.
... Student teachers are expected to develop PCK in their classrooms. Livy and Downton [16] recommended that fieldbased experiences should develop the student teachers' PCK for teaching that includes how to recognize and support students' emerging reasoning skills. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. ...
... [17] found that student teachers had underdeveloped PCK because the construct of teaching and learning was complex. Additionally, Livy and Downton [16] also indicated that student teachers had problems with instructional design, including teaching methods, techniques and measurement and evaluation methods [18]. The problems regarding the insufficient PCK may cause student teachers to seek strategies for improvement. ...
... As for PCK that involves a complex and different situation [16], online information exchange through social media may be insufficient. Even though studies indicated that student teachers' PCK is underdeveloped [17], the way of interchange field-based experience on social media were not enough to promote their PCK. ...
According to literature review, student teachers often encounter problems regarding teaching-related practices, anxiety on practicum, and interpersonal relationships with school personnel. Social media can provide information and support. The purpose of this study is to investigate what social media can benefit student teachers in. A Facebook Group for Taiwanese interns was built. A quasi-experimental pre- and post-test control group design was employed among student teachers. The analysis of the validated questionnaires consisted of 105 in E-Group and 40 in C-Group. This study concludes that social media use benefited student teachers in decrease of teacher anxiety on practicum; however, the use cannot promote their pedagogical content knowledge, neither strengthen their interpersonal interaction with school personal.
... The universities as a provider of teacher education program should prepare their student teacher who will afford high quality on teaching by accommodating a high quality practicum on teaching. Preparing good quality pre-service teacher can be a key to create excellent teacher in the future [2]. Collaboration between preservice teacher and in-service teacher will head to achieve new goals, tools in teaching, skills, and train pre-service teacher to cross their own boundaries to work with others and create their own new solution for every problem found [3]. ...
... SEA-Teacher Project accommodates student teachers in South East area to practice and develop their teaching skills with challenging ambience. The goals of this program are : (1) to enable pre-service student teachers to develop their teaching skills and pedagogy, (2) to encourage the pre-service student teachers to practice their English skills, (3) to allow the pre-service student teachers to gain a broader regional and world view, and (4) to expose future teachers to diverse teaching and learning situations and opportunities, and the value of flexibility [5]. Pre-service Chemistry student teacher is one of student teachers embarked in this program besides Biology, Mathematics, Physics, English, Early Childhood, and Primary School student teachers. ...
South East Asia is undergoing massive development in many sectors, so do the education. Good education needs a high quality teacher. SEA Teacher Project is a project held by SEAMEO to revitalize teacher education. Practicum for pre-service teachers plays a big role in the development of teacher in the future. Departing from SEA Teacher Project’s goals, this case study research aimed to explore how the teachers in the practice schools provided analogies and used visual aids in chemistry learning and how were student teachers’ perspectives in chemistry learning. The participants were 4 chemistry pre-service student teachers who experienced this program between 2016 and 2019. This research also explored the issues and challenges in the field to improve chemistry pre-service student teachers’ preparation, the fruitfulness of the program based on pre-service student teachers’ perspective. The results showed that analogies and visual aids are strengthen each other in chemistry learning. The SEA Teacher Project held by 2016 to 2019 has given significant experience of teaching for the participants.
... It is an essential pedagogical approach that involves noticing how students responded during the lesson phases and how the lesson structure promoted and accommodated student learning. In other words, teachers can reinforce the point of choosing and selecting students to share their strategies (Livy & Downton, 2018). Moreover, as described by Lee and Lee (2023), teachers should employ differentiated instruction to accommodate diverse learning styles and levels of prior knowledge and support struggling students. ...
This study seeks to characterize the Technological Pedagogical Content Knowledge of novice lower secondary mathematics teachers who do and do not possess an educator certificate based on the components of Technological Knowledge (TK), Pedagogical Knowledge (PK), Content Knowledge (CK), Technological Content Knowledge (TCK), Pedagogical Content Knowledge (PCK), Technological Pedagogical Knowledge (TPK), and Technological Pedagogical Content Knowledge (TPACK). This research employed a case study approach conducted in two secondary schools in Merauke Regency. Two secondary mathematics teachers with a bachelor's degree in mathematics education and five years of teaching experience were recruited as participants. We collected data from learning classroom observation guidelines, learning-practice interviews, and task-based interviews. Findings show that certified teachers can implement learning according to the lesson plans that have been designed and can teach and answer questions about the assigned material effectively. Meanwhile, the uncertified teacher performs excellently using technology, such as proficiency with WhatsApp, Zoom, Microsoft OneNote, Microsoft PowerPoint, learning videos, and projectors. In contrast, regarding pedagogical and material abilities, the non-certified teachers have yet to be able to apply lesson plans to learning and continue to struggle to answer some predetermined questions.
... Meanwhile, a study by Höhn et al. (2017) also conducted on 444 Germans, also found that the PUB factor affects pre-service teachers' mastery of knowledge. A recent study conducted by Livy & Downton (2018) for teachers in Australia also found that PUB factors affect teachers' PMUP. In addition, a study by Guo et al. (2020) also shows that there is a significant relationship between PUB and PMUP. ...
This study investigates the effect of learning opportunities and beliefs on the efficacy of teaching mathematics on knowledge of mathematics for teaching. Using a structured questionnaire along with paper and pencil tests adapted from the literature review, data were collected from 187 prospective primary school teachers at the Indonesian Teacher Education Institute. Data were analyzed using Smart PLS 3.0. The results of the structural equation model show that both opportunities to learn-Practicum (β = 0.395, p < 0.001) and opportunities to learn -Program (β = 0.324, p < 0.001) are positively related to mathematics. The second regression analysis is to examine the influence of teaching efficacy beliefs on mathematics. The results showed that beliefs about expectations of learning outcomes in mathematics (β = 0.322, p < 0.001) were positively related to mathematics, while personal mathematics teaching efficacy beliefs (β = 0.017, p > 0.1) were not related to mathematics. Overall, the confidence factor and opportunities to learn explain the total 54% variance in mathematics. The implications of these findings for the successful implementation of teacher education programs in Indonesia are further elaborated.
... However, existing studies mainly investigated their geometry content knowledge (Browning et al., 2014;Fujita, 2012;Menon, 1998;Pickreign, 2007;Reinke, 1997). In addition, studies that measured pre-service elementary teachers' both geometry content and pedagogical content knowledge comprehensively and reliably are relatively limited (Livy & Downton, 2018;Martinovic & Manizade, 2018;Robichaux-Davis & Guarino, 2016). Therefore, this study is developed to identify geometry content and pedagogical content knowledge for teaching 2dimensional (2D) shapes at the elementary level through a literature review and to evaluate the reliability and validity of the GKT-2D scale, an instrument specifically designed to measure such knowledge. ...
... Since previous research has indicated that some primary teachers hold a fragmented understanding of reasoning (Herbert et al., 2015)) prompting an increased focus on professional learning. A range of approaches have been trialled to enhance teachers' understanding and teaching of mathematical reasoning, such as demonstration lessons Herbert, Vale, Bragg, Loong, & Widjaja, 2015;Livy & Downton, 2018); surveys of teachers' use of reasoning language (Clarke, Clarke & Sullivan, 2012); peer-learning-teams (Herbert & Bragg, 2021;Herbert & Bragg, 2020) and professional learning workshops (Hilton, Hilton, Dole & Goos, 2016). ...
Mathematical reasoning is crucial for a deep understanding of mathematics, yet many primary teachers have a limited conception of reasoning. A deeper knowledge of reasoning is required to notice, teach, and assess it effectively. Sixteen pairs of teachers from four primary schools in Victoria, Australia, engaged in a professional learning program intended to assist teachers embed reasoning in their mathematics lessons. Transcripts of post lesson discussions were examined to investigate any changes in their knowledge of reasoning. Analysis indicates that most pairs demonstrated increased use of reasoning language. This study reveals the efficacy of assessing students’ reasoning using the Assessing Mathematical Reasoning Rubric (AMRR) for building teachers’ knowledge of reasoning. Examples of three pairs of teachers are provided to illustrate teachers’ growing awareness of aspects of reasoning.
... Moreover, there is evidence that exposure to pedagogies that view productive struggle as integral to mathematics learning, such as problem-based approaches to learning mathematics, can further enhance teacher attitudes. Livy and Downton presented evidence that pre-service teachers can become more aware of the value of student struggle through exposure to a lesson structure that facilitates student exploration of a problem prior to instruction [13]. Likewise, Russo et al. found that teachers who had been exposed to professional learning around such problem-based approaches to learning mathematics had more positive attitudes towards struggle than teachers who had not participated in such professional learning [12]. ...
Given what is known about the importance of productive struggle for supporting student learning of mathematics at all levels, the current study sought to examine teacher attitudes towards student struggle when students learn mathematics in remote learning settings compared with classroom settings. Eighty-two Australian early years primary teachers involved in a professional learning initiative focused on teaching mathematics through sequences of challenging tasks completed a questionnaire inviting them to compare the two settings. Drawing on a mixed-methods approach, we found that teachers were more positive about the value of student struggle in classroom-based settings compared with remote learning settings. Qualitative analysis of open-ended responses revealed four themes capturing why teachers viewed efforts to support productive struggle in a remote learning setting as potentially problematic: absence of a teacher-facilitated, synchronous, learning environment; parents’ negative attitudes towards struggle when learning mathematics; lack of social connection and peer-to-peer collaboration; and difficulties accessing learning materials. Suggestions for mitigating some of these challenges in the future are put forward.
... Some studies (e.g. Livy & Downton, 2018;Stephens, 2008;Sullivan, 2018;Tchoshanov, Quinones, Shakirova, Ibragimova, & Shakirova, 2017) suggest that preservice teachers need to have conceptual knowledge for teaching algebra. Teachers need to have knowledge of using multiple representations, justifying reasoning, using generalization and posing problems for teaching algebraic reasoning (Bair & Rich, 2011). ...
Teachers use their content and pedagogical content knowledge for teaching algebra. For this reason, the examination of how teachers use this knowledge may help shed light on how students learn algebra, especially in determining why they usually have difficulties. The aim of the current study is to reveal what teachers know, and propose what they actually need to know for teaching the simplification and equivalence of algebraic expressions. The multiple-case study design was used for this study to compare and contrast the two middle school teachers" lesson planning and instruction. The data corpus included lesson plans, actual instruction records, and post-observation interviews. Data analysis was conducted using the Mathematical Knowledge for Teaching (MKT) model. The findings indicated that both teachers had a lack of specialized content knowledge about mathematical representations such as algebra tiles. They did not use algebra tiles effectively and could not link algebraic and geometric representations that underlie the idea of multiplication. It was observed that both teachers generally used unknowns and variables interchangeably indicating the inadequacy of their common content knowledge. In the planning process, the two teachers were able to state the common misconceptions that the students generally had and the ways of addressing them. Through the cases of these two teachers, it was observed that teachers need to have a good conceptual mathematical understanding and also knowledge of students" thinking in order to design effective lessons. Based on the findings, the types of knowledge that the teachers need to have are outlined and the theoretical and practical implications of the study are discussed.
... Some studies (e.g. Livy & Downton, 2018;Stephens, 2008;Sullivan, 2018;Tchoshanov, Quinones, Shakirova, Ibragimova, & Shakirova, 2017) suggest that preservice teachers need to have conceptual knowledge for teaching algebra. Teachers need to have knowledge of using multiple representations, justifying reasoning, using generalization and posing problems for teaching algebraic reasoning (Bair & Rich, 2011). ...
Teachers use their content and pedagogical content knowledge for teaching algebra. For this reason, the examination of how teachers use this knowledge may help shed light on how students learn algebra, especially in determining why they usually have difficulties. The aim of the current study is to reveal what teachers know, and propose what they actually need to know for teaching the simplification and equivalence of algebraic expressions. The multiple-case study design was used for this study to compare and contrast the two middle school teachers" lesson planning and instruction. The data corpus included lesson plans, actual instruction records, and post-observation interviews. Data analysis was conducted using the Mathematical Knowledge for Teaching (MKT) model. The findings indicated that both teachers had a lack of specialized content knowledge about mathematical representations such as algebra tiles. They did not use algebra tiles effectively and could not link algebraic and geometric representations that underlie the idea of multiplication. It was observed that both teachers generally used unknowns and variables interchangeably indicating the inadequacy of their common content knowledge. In the planning process, the two teachers were able to state the common misconceptions that the students generally had and the ways of addressing them. Through the cases of these two teachers, it was observed that teachers need to have a good conceptual mathematical understanding and also knowledge of students" thinking in order to design effective lessons. Based on the findings, the types of knowledge that the teachers need to have are outlined and the theoretical and practical implications of the study are discussed.
... La tarea problemática se asigna posteriormente a los estudiantes, quienes deben formularla y reformularla para resolverla. Con esto, no planteamos que los futuros maestros deban necesariamente experimentar con estudiantes de escolaridad primaria, aunque estudios previos hayan señalado su potencialidad (ver Crespo, 2003;Livy y Downton, 2018). ...
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.
... Algunos enfoques que permiten a los estudiantes interactuar con estudiantes reales (e.g. Crespo, 2003;Livy y Downton, 2017), permite a los futuros docentes ser conscientes de las capacidades de sus posibles futuros estudiantes. ...
En este trabajo presentamos un análisis del conocimiento sobre el aprendizaje de la resolución de problemas, específicamente sobre el conocimiento de los estudiantes como resolutores, manifestado por 149 futuros profesores de educación primaria al terminar su formación. A través de un cuestionario los sujetos manifestaron un mayor conocimiento sobre características de resolutores exitosos. Por ejemplo, reconocen que los buenos resolutores presentan una mejor organización de su conocimiento y un adecuado manejo de las emociones. Sin embargo, muestran respuestas dudosas al pedirles identificar características de resolutores novatos como la identificación de información importante o el uso de estrategias inadecuadas.
... This clearly shows that OTL is an important factor affecting the mastery of knowledge and academic achievement of future teachers. In addition, recent studies conducted by Livy and Downton (2018) also found that the OTL-Practicum factor has influenced the mastery of mathematical knowledge for teaching among pre-service teachers. The findings from the case study conducted among 20 second-year pre-service teacher found that the course experiences provided an opportunity to extend pre-service teachers' knowledge. ...
Issues about low level of mathematical knowledge for teaching among pre-service teachers has raised the question on the effectiveness of the mathematics teacher education program which has been planned and implemented by the Malaysian Institute of Teacher Education (MITE). This study was conducted to identify factors that affect mathematical knowledge for teaching (MKT) among pre-service teachers in Institute of Teacher Education (ITE). The influence of mathematical belief and opportunity to learn (OTL) have been tested to explain the factors affecting MKT. Using a structured questionnaire together with paper and pencil test adapted from the literature reviewed, data were collected from 105 pre-service teachers in MITE. Data were analysed using SmartPLS version 3.0. The result of the structural equation model indicated that OTL-Practicum (β=0.491, p<0.001) and OTL-Program (β=0.368, p<0.001) has a positive relationship with mathematical knowledge for teaching. Besides that, the result for the impact of OTL on mathematical belief, it showed that OTL-Practicum (β=0.208, p<0.001) and OTL-Program (β=0.243, p<0.001) has a positive relationship with constructivist belief, whereas OTL-Program was negatively related to traditional belief (β=-0.283, p<0.001). Overall, the model explained 53.9% of the variance in mathematical knowledge for teaching. Implications from these findings to the ITE were further elaborated.
... Kazemi et al. (2009) recommended that PSTs must be encouraged to rehearse their teaching if they are to enact ambitious mathematics instruction. Helping PSTs to make connections with their teacher identity and what they learn at university and teaching may also include observation of model lessons (Livy and Downton 2018) or discussion of student work samples (Livy et al. 2017). Grootenboer (2008) reminded us that learning to teach also involves emotions, attitudes and beliefs and that teacher educators should not ignore these if PTSs are to adopt new mathematical beliefs. ...
There is a consensus that we need to improve the quality of pre-service teacher education, and teachers’ mathematical content knowledge is critical for teaching. Identifying opportunities and influences that assist pre-service teachers to extend their mathematical content knowledge throughout their teacher education programme is important. This paper reports on qualitative data, collected over 4 years from two typical pre-service teachers whose developing mathematical content knowledge was investigated during their primary and secondary programme. These data were analysed and reported using the four dimensions of the Knowledge Quartet: foundation knowledge, transformation, connection and contingency. The results highlight the consequences of programme structure in order to help pre-service teachers to establish and sustain a positive mathematics learner identity, build teacher identity and develop mathematical content knowledge.
In our mathematics methods courses for elementary preservice teachers, we work to uncover and confront students' understandings as well as misconceptions about important mathematical topics. Karp and colleagues' ( Teaching Children Mathematics, 21(1), 18–25) “13 Rules That Expire” article has been a useful resource for us to highlight and challenge students' mathematical misconceptions. In this article, we share the components of an iteratively developed assignment focused on the 13 rules article. We detail how students explored the “lives” of these rules and investigated alternatives to pervasive mathematics shortcuts and tricks. We posit that our intentionally designed assignment provides an opportunity for students to think critically about their views, beliefs, and misconceptions, which encourages a mindset for preservice teachers to consider the kinds of mathematical misconceptions they will encounter in their classroom teaching. We anticipate that sharing this developed assignment with other mathematics teacher educators could provide opportunities for additional adaptations in unique contexts and continued improvements to support future teachers.
Whilst spatial reasoning skills have been found to predict mathematical achievement, little is known about how primary (elementary) students’ conceptual understanding of three-dimensional objects develops. In this article, we report a qualitative study and the impact of rich learning experiences on 48 Years 3–6 students’ geometric reasoning relating to prisms. A one-to-one task-based interview, refined by the researchers, was used to assess student learning. Coding and data analysis were informed by our previous research. The findings reveal noticeable shifts in students’ knowledge of and reasoning about prisms, their ability to construct and describe prisms with geometric language, and their visualisation and spatial structuring skills. The implications of these findings highlight the importance of teachers’ choice of tasks that require students to compose and decompose three-dimensional (3D) objects; compare 3D objects through physical and mental transformations; take different perspectives; and visualise and reason geometrically.
The present study investigates the influence of opportunities to learn (OTL) and mathematics teaching efficacy beliefs (MTEB) towards mathematical knowledge for teaching (MKT). Using a structured questionnaire together with paper and pencil test adapted from the literature reviewed, data were collected from 187 pre-service elementary teachers in Institute of Teacher Education (ITE) Malaysia. Data were analysed using SmartPLS 3.0. The result of the structural equation model indicated that both OTLPracticum (β = 0.395, p < 0.001) and OTL-Program (β = 0.324, p < 0.001) was positively related to MKT. The second regression analysis was to examine the impact of mathematics teaching efficacy belief on the MKT. The results showed that mathematics teaching outcome expectancy belief (MTOEB) (β = 0.322, p < 0.001) was positively related to MKT, whereas personal mathematics teaching efficacy belief (β = 0.017, p > 0.1) was not related to MKT. Overall, the belief and OTL factors explaining a total of 54% variance in MKT. Implications from these findings to the successful teacher’s education program implementation in Malaysia were further elaborated.
This article presents a systematic literature review about disciplinary expert teachers in primary science and mathematics education. This is a timely synthesis of the literature, as current reforms in teacher education in Australia and internationally require primary teachers to have specialised knowledge in a learning area. Systematic review protocols were used to identify and evaluate the relevance of numerous articles of which thirty-seven were included in the final analysis. Findings show insufficient evidence about whether expert teachers have a positive impact on instructional quality and student learning. Implications are discussed with reference to the current policy moment in Australia and teacher education more broadly.
While some teacher educators work alongside teachers with their pre-service teachers in school settings, it is less common for teachers to work alongside teacher educators in university settings. The following chapter provides insights on how a mathematics lecturer and a Year 1 teacher taught primary pre-service teachers in a university classroom. Narratives from the co-teachers and two colleagues who observed lessons and interviewed the pre-service teachers tell the story of a co-teaching experience which developed relationships, a sense of community, and diverse skills and expertise of the co-teachers, along with the impact that the experience had on the participants.
The following is a report of an exploration of what mathematical reasoning might look like
in classrooms. Focusing on just one lesson in one classroom, data are presented that indicate
that upper primary students are willing and able to reason for themselves, especially in
classrooms in which the culture for such reasoning has been established. It seems that the
opportunities to reason are a product of the tasks that are posed, the structuring of the
classroom, and the willingness of the teachers to allow students to engage with the tasks for
themselves.
This theoretical paper argues that the reform in mathematics towards more problem-based learning can be made consistent with cognitive load theory through the use of carefully designed challenging tasks. It is argued that such tasks can provide the benefits of problem-based approaches whilst being cognisant of the issue of cognitive overload. Possible directions for future research are suggested.
This paper outlines the new opportunities that that will be changing the landscape of geometry education at the primary school level. These include: the research on spatial reasoning and its connection to school mathematics in general and school geometry in particular; the function of drawing in the construction of geometric meaning; the role of digital technologies; the importance of transformational geometry in the curriculum (including symmetry as well as the isometries); and, the possibility of extending primary school geometry from its typical emphasis on vocabulary (naming and sorting shapes by properties) to working on the composing/decomposing, classifying, comparing and mentally manipulating both two- and three-dimensional figures. We discuss these opportunities in the context of historical developments in the nature and relevance of school geometry. The aim is to motivate and connect the set of papers in this special issue.
The following is a report on an investigation into ways of supporting teachers in converting challenging mathematics tasks into classroom lessons and supporting students in engaging with those tasks. Groups of primary and secondary teachers, respectively, were provided with documentation of ten lessons built around challenging tasks. Teachers responded to survey items in both Likert and free format style after teaching the ten lessons. The responses of the teacher participants indicate that the lesson structure we proposed was helpful, and the elements of the lessons suggested to teachers were both necessary and sufficient for supporting students in engaging with the challenging tasks. Implications for teacher educators and curriculum developers are offered.
While teacher content knowledge is crucially important to the improvement of teaching and learning, attention to its development and study has been uneven. Historically, researchers have focused on many aspects of teaching, but more often than not scant attention has been given to how teachers need to understand the subjects they teach. Further, when researchers, educators and policy makers have turned attention to teacher subject matter knowledge the assumption has often been that advanced study in the subject is what matters. Debates have focused on how much preparation teachers need in the content strands rather than on what type of content they need to learn. In the mid-1980s, a major breakthrough initiated a new wave of interest in the conceptualization of teacher content knowledge. In his 1985 AERA presidential address, Lee Shulman identified a special domain of teacher knowledge, which he referred to as pedagogical content knowledge. He distinguished between content as it is studied and learned in disciplinary settings and the "special amalgam of content and pedagogy" needed for teaching the subject. These ideas had a major impact on the research community, immediately focusing attention on the foundational importance of content knowledge in teaching and on pedagogical content knowledge in particular. This paper provides a brief overview of research on content knowledge and pedagogical content knowledge, describes how we have approached the problem, and reports on our efforts to define the domain of mathematical knowledge for teaching and to refine its sub- domains.
This article focuses on the Problem-Solving Cycle (PSC), a model of professional development designed to assist teachers in supporting their students' mathematical reasoning. Each PSC is a series of three interrelated workshops in which teachers share a common mathematical and pedagogical experience, organized around a rich mathematical task. Throughout the workshops, teachers delve deeply into issues involving mathematical content, pedagogy, and student thinking as they pertain to the selected task. We analyze this professional development model in relation to the ways it supports the development of content and pedagogical content knowledge. We highlight the ways in which specific knowledge strands are foregrounded during each of the three PSC workshops, while also demonstrating their interconnectedness.The improvement of students' opportunities to learn mathematics depends fundamentally on teachers' skill and knowledge. No curriculum or framework is self-enacting, nostudents self-teaching. Moreover, teachers are often expected to teach mathematical topics and skills in ways substantially different from the ways in which they themselves learned that content.… Hence, if students' learning is to improve, teachers' professional learning opportunities are key. (Boaler & Humphreys, 20053.
Boaler , J. and
Humphreys , C. 2005. Connecting mathematical ideas: Middle school video cases to support teaching and learning., Portsmouth, NH: Heinemann. View all references)
The teaching and learning of Geometry has been identified in much of the literature as being problematic and the mathematics strand where many teachers feel least knowledgeable and least confident to teach. This paper describes a school-based project which sought to develop teacher knowledge and confidence in this strand via the use of Professional Learning Communities (DuFour & Reeves, 2016) and Instructional Coaching.
This book helps readers to become better, more confident teachers of mathematics by enabling them to focus critically on what they know and what they do in the classroom. Building on their close observation of primary mathematics classrooms, the authors provide those starting out in the teaching profession with a four-stage framework which acts as a tool of support for developing their teaching: Making sense of foundation knowledge – focusing on what teachers know about mathematics; Transforming knowledge – representing mathematics to learners through examples, analogies, illustrations, and demonstrations; Connection – helping learners to make sense of mathematics through understanding how ideas and concepts are linked to each other; Contingency – what to do when the unexpected happens Each chapter includes practical activities, lesson descriptions, and extracts of classroom transcripts to help teachers reflect on effective practice. Video versions of these lessons are also available on a companion website. © Tim Rowland, Fay Turner, Anne Thwaites and Peter Huckstep 2009.
Prior studies suggest that struggling to make sense of mathematics is a necessary component of learning mathematics with understanding. Little research exists, however, on what the struggles look like for middle school students and how they can be productive. This exploratory case study, which used episodes as units of analysis, examined 186 episodes of struggles in middle school students as they engaged in tasks focused on proportional reasoning. The study developed a classification structure for student struggles and teacher responses with descriptions of the kinds of student struggle and kinds of teacher responses that occurred. The study also identified and characterized ways in which teaching supported the struggles productively. Interaction resolutions were viewed through the lens of (a) how the cognitive demand of the task was maintained, (b) how student struggle was addressed and (c) how student thinking was supported. A Productive Struggle Framework was developed to capture the episodes of struggle episodes from initiation, to interaction and to resolution. Data included transcripts from 39 class session videotapes, teacher and student interviews and field notes. Participants were 327 6th- and 7th-grade students and their six teachers from three middle schools located in mid-size Texas cities. This study suggests the productive role student struggle can play in supporting ‘‘doing mathematics’’ and its implications on student learning with understanding. Teachers and instructional designers can use this framework as a tool to integrate student struggle into tasks and instructional practices rather than avoid or prevent struggle.
Lee S. Shulman builds his foundation for teaching reform on an idea of teaching that emphasizes comprehension and reasoning, transformation and reflection. "This emphasis is justified," he writes, "by the resoluteness with which research and policy have so blatantly ignored those aspects of teaching in the past." To articulate and justify this conception, Shulman responds to four questions: What are the sources of the knowledge base for teaching? In what terms can these sources be conceptualized? What are the processes of pedagogical reasoning and action? and What are the implications for teaching policy and educational reform? The answers — informed by philosophy, psychology, and a growing body of casework based on young and experienced practitioners — go far beyond current reform assumptions and initiatives. The outcome for educational practitioners, scholars, and policymakers is a major redirection in how teaching is to be understood and teachers are to be trained and evaluated.
This article was selected for the November 1986 special issue on "Teachers, Teaching, and Teacher Education," but appears here because of the exigencies of publishing.
Top drawer teachers: Geometric reasoning
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Aspects of teachers' pedagogical content knowledge for decimals
- H L Chick
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Chick, H. L., Baker, M., Pham, T., & Cheng, H. (2006). Aspects of teachers' pedagogical content knowledge for decimals. In J. Novotna, H. Moraova, M. Kraka, & N.
Stehlikova (Vol. Eds.), Proceedings 30th Conference of the International Group for the Psychology of Mathematics Education: Vol. 2, (pp. 297-304).
A highly capable Year 6 student's response to a challenging mathematical task
- S Livy
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- P Sullivan
Livy, S., Ingram, N., Holmes, M., Linsell, C., & Sullivan, P. (2016). A highly capable Year 6 student's response to a challenging mathematical task. In B. White, M.
Chinnappan, & S. Trenholm (Eds.). Opening up mathematics education research. Proceedings of the 39th annual conference of the Mathematics Education Research Group
of Australasia (pp. 665-668).
Principles and standards for school mathematics
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Mueller, M., Yankelewitz, D., & Maher, C. (2014). Teachers promoting student mathematical reasoning. Investigations in Mathematics Learning, 7(2), 1-20.
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Teachers holding back from telling: A key to student persistence on challenging tasks
- A Roche
- D Clarke
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19(4), 3-8.
Planning and teaching mathematics lessons as a dynamic, interactive process
- P Sullivan
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Sullivan, P., Zevenbergen, R., & Mousley, J. (2005). Planning and teaching mathematics lessons as a dynamic, interactive process. In H. L. Chick, & J. L. Vincent (Vol.
Eds.), Proceedings of the 29th conference of the International Group for the Psychology of Mathematics Education: Vol. 4, (pp. 249-256).
Classroom culture, challenging mathematical tasks and student persistence
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Sullivan, P., Aulert, A., Lehmann, A., Hislop, B., Shepherd, O., & Stubbs, A. (2013). Classroom culture, challenging mathematical tasks and student persistence. In V.
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Teaching mathematics: Using research informed strategies Australian education review
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