Quality issue in the design and use of resources by mathematics teachers

The profusion of resources, mostly on the Internet, makes the issue of their quality more and more pressing. Teachers often find themselves unable to choose from among them those that would be the most relevant to their educational goal and to the context of their classes. In our previous work, we claimed that acquiring resource analysis skills is one of the keys to the teachers’ professional development supporting the integration of dynamic geometry systems. Based on this assumption, we have designed a quality questionnaire for the i2geo repository aiming at framing the analysis of available resources by the platform users. The research reported in this paper, which is a continuation of this work, explores the way the quality criteria are taken into account in the design and use of resources by members of the community that has emerged around the repository and sheds light on possible evolutions of teachers’ and teacher educators’ professional practices due to their involvement in the resource quality evaluation.     

Adapting Japanese Lesson Study to enhance the teaching and learning of geometry and spatial reasoning in early years classrooms: a case study

Increased efforts are needed to meet the demand for high quality mathematics in early years classrooms. Despite the foundational role of geometry and spatial reasoning for later mathematics success, the strand receives inadequate instructional time and is limited to concepts of static geometry. Moreover, early years teachers typically lack both content knowledge and confidence in teaching geometry and spatial reasoning. We describe our attempt to deal with these issues through a research initiative known as the Math for Young Children project. The project integrates effective features of both design research and Japanese Lesson Study and is designed to support teachers in developing content knowledge and new approaches for teaching geometry and spatial reasoning. Central to our Professional Development model is the integration of four adaptations to the Japanese Lesson Study model: (1) teachers engaging in the mathematics, (2) teachers designing and conducting task-based clinical interviews, (3) teachers and researchers co-designing and carrying out exploratory lessons and activities, and (4) the creation of resources for other educators. We present our methods and the results of our adaptations through a case study of one Professional Learning Team. Our results suggest that the adaptations were effective in: (1) supporting teachers’ content knowledge of and comfort level with geometry and spatial reasoning, (2) increasing teachers’ perceptions of young children’s mathematical competencies, (3) increasing teachers’ awareness and commitment for the inclusion of high quality geometry and spatial reasoning as a critical component of early years mathematics, and (4) the creation of innovative resources for other educators. We conclude with theoretical considerations and implications of our results.     

A Telecommunications Project to Empower Kansas Elementary/Middle Level Teachers as Change Agents for Integrated Science and Mathematics Education

Nineteen elementary and 5 middle school in-service teachers served as regional lead teachers to provide 13 two- and four-day workshops for 235 in-service teachers. The lead teachers attended 100 hours of inservice training on national science/math reform curricula, cognitive theory, pedagogical strategies for enhancing problem solving, and authentic assessment procedures. The lead teachers then selected $30,000 worth of instructional materials for use in their classrooms. Subsequently, they developed four hands-on-manipulative-based workshops through extensive use of video-teleconferencing technology. The lead teachers strongly believe that this extensive project has made a permanent positive impact on the quality and quantity of integrated science and math instruction in Kansas.     

Prompting Teacher Geometric Reasoning through Coaching in a Dynamic Geometry Software Context

This study investigated the ways in which four middle grades teachers developed mathematical knowledge for teaching (MKT) geometry as they implemented dynamic geometry software in their classrooms with the assistance of a coach. Teachers developed various components of MKT by observing coaches teach, by dynamic discourse with students, which is discourse with respect to dynamic geometry software images, and by discussions with coaches. The dynamic geometry software environment proved productive as coaches guided teachers’ growth in explanations, examples, and definitions. The environment also helped teachers discover unnoticed abilities among their low achievers. Moreover, teachers developed confidence to teach with more expert uses of dynamic geometry software.     

Re-use? Is this re-use?

An initiative such as the Intergeo project has the clear mission to allow many teachers to gain easy access to resources that can support their teaching practice when using them. Re-use then appears when the sole usage is insufficient and needs adaptations to the specific context. This paper investigates the common practices of re-use in computer science literature, describes how it was implemented in the i2geo platform, and how the actual usage of the platform shows that the definition we first envisioned and that is widespread is short sighted. Following the observations of the practices, generalizations of the notion of re-use are proposed.     

The Impact of Problem Posing on Elementary Teachers' Beliefs About Mathematics and Mathematics Teaching

This study investigated the impact of incorporating problem posing in elementary classrooms on the beliefs held by elementary teachers about mathematics and mathematics teaching. Teachers participated in a year‐long staff development project aimed at facilitating the incorporation of problem posing into their classrooms. Beliefs were examined via pre‐ and postsurvey. Results indicated a positive impact on their beliefs about mathematics and mathematics instruction. Data from open‐ended written responses verified the impact of problem posing on the teachers and their classrooms. Based on these findings, it is recommended that problem posing be incorporated into all professional learning and undergraduate education programs.     

Analyzing Student Work as a Professional Development Activity

When worthwhile mathematical tasks are used in classrooms, they should also become a crucial element of assessment. For teachers, using these tasks in classrooms requires a different way to analyze student thinking than the traditional assessment model. Looking carefully at students' written work on worthwhile mathematical tasks and listening carefully while students explore these worthwhile tasks can contribute to a teacher's professional development. This paper reports on a professional development activity in which teachers analyzed mathematical tasks, predicted students' achievement on tasks, evaluated students' written work, listened to students' reasoning, and assessed students' understanding. Teachers' engagement in this way can help them develop flexibility and proficiency in the evaluation of their own students' work. These experiences allow teachers the opportunity to recognize students' potential, strengthen their own mathematical understanding, and engage in conversations with peers about assessment and instruction.     

The didactical tetrahedron as a heuristic for analysing the incorporation of digital technologies into classroom practice in support of investigative approaches to teaching mathematics

There have been various proposals to expand the heuristic device of the didactical triangle to form a didactical tetrahedron by adding a fourth vertex to acknowledge the significant role of technology in mediating relations between content, student and teacher. Under such a heuristic, the technology vertex can be interpreted at several levels from that of the material resources present in the classroom to that of the fundamental machinery of schooling itself. At the first level, recent research into teacher thinking and teaching practice involving use of digital technologies indicates that, while many teachers see particular tools and resources as supporting the classroom viability of investigative approaches to mathematics, the practical expressions of this idea in lessons vary in the degree of emphasis they give to a didactic of reconstruction of knowledge, as against reproduction. At the final level, examining key structuring features of teaching practice makes clear the scope and scale of the situational adaptation and professional learning required for teachers to successfully incorporate use of digital tools and resources in support of investigative approaches. These issues are illustrated through examining contrasting cases of classroom use of dynamic geometry in professionally well-regarded mathematics departments in English secondary schools.     

Teacher preparation for a problem-solving curriculum in Singapore

Problem solving is at the heart of the Singapore Mathematics curriculum. However, it remains a challenge for teachers to realise this curricular goal in practice. Here, we review the efforts of Singapore mathematics teacher educators in incorporating problem-solving (teaching) competency in teacher education and PD programmes. We discuss conceptual and practical issues, actions taken and changes made in building teachers?? capacity to enact a problem-solving curriculum in a school-based design experiment project. In the project, teachers learnt problem solving, observed and then carried out lessons, using the ??Mathematics Practical????akin to the science practical??as key to instruction and assessment.     

Developing teachers' models for assessing students' competence in mathematical modelling through lesson study

Applications and modelling have gained a prominent role in mathematics education reform documents and curricula. Thus, there is a growing need for studies focusing on the effective use of mathematical modelling in classrooms. Assessment is an integral part of using modelling activities in classrooms, since it allows teachers to identify and manage problems that arise in various stages of the modelling process. However, teachers’ difficulties in assessing student modelling work are a challenge to be considered when implementing modelling in the classroom. Thus, the purpose of this study was to investigate how teachers’ knowledge on generating assessment criteria for assessing student competence in mathematical modelling evolved through a professional development programme, which is based on a lesson study approach and modelling perspective. The data was collected with four teachers from two public high schools over a five-month period. The professional development programme included a cyclical process, with each cycle consisting of an introductory meeting, the implementation of a model-eliciting activity with students, and a follow-up meeting. The results showed that the professional development programme contributed to teachers’ knowledge for generating assessment criteria on the products, and the observable actions that affect the modelling cycle.     

Special Issue Article: A Look at Technology's Role in Professional Development of Mathematics Teachers at the Middle School Level

In last month's issue of School Science and Mathematics, Glenda Lappan addressed the dilemma of supporting teachers in continuing to grow professionally in four domains: Learning more mathematics content; improving their pedagogical skills; improving assessment skills; and addapting the curriculum to the needs of their students. Though it is clear that this kind of learning is best done in the context of what is going on in the classroom, teachers' schedules and other problems make it difficult to implement a coherent classroom based, professional development program. In this article, two more knowledge domains are added to this mix: the ability to (a) use technology and (b) teach with technology effectively. Since the process of learning and teaching is a dynamic one, the author describes his vision of the classroom as a laboratory where teachers get to practice and improve in these six areas and get feedback from an audience of their peers. Reflections are based on a current project in Paterson, New Jersey where the author helps middle school teachers use computer software to improve their mathematics teaching and learning.     

Pre-constructed dynamic geometry materials in the classroom – how do they facilitate the learning of ‘Similar Triangles’?

The use of dynamic geometry software (DGS) is becoming increasingly familiar among teachers, but letting students conduct inquiries using computers is still not a welcome idea. In addition to logistics and discipline concerns, many teachers believe that mathematics at the lower secondary level can be learned efficiently through practice alone. Thus, the application of DGS remains limited to demonstration and explanation. This article discusses how a set of pre-constructed dynamic geometry (DG) materials was designed to teach the ‘similar triangles’ concept. The reactions and behaviour of students with relatively low levels of mathematic achievement are also analysed. Finally, the potential value of pre-constructed DG materials, with lab sheets and teacher intervention, in inquiry activities for junior-level students is discussed.     

Using resource scarceness characteristics to solve the multi-mode resource-constrained project scheduling problem

In the past decades, resource parameters have been introduced in project scheduling literature to measure the scarceness of resources of a project instance. In this paper, we incorporate these resource scarceness parameters in the search process to solve the multi-mode resource constrained project scheduling problem, in which multiple execution modes are available for each activity in the project. Therefore, we propose a scatter search algorithm, which is executed with different improvement methods, each tailored to the specific characteristics of different renewable and nonrenewable resource scarceness values. Computational results prove the effectiveness of the improvement methods and reveal that the procedure is among the best performing competitive algorithms in the open literature.     

多阶段并发的协同物流网络任务-资源集成调配模型

协同物流网络作为动态开放的网络需要整合多种资源,以完成多项并行的物流任务。为此,本文从物流服务集成商的角度,研究了带时间窗和资源约束的任务—资源集成调配问题。通过将物流任务分解为不同阶段,构造多阶段并发的协同物流网络任务—资源集成调配模型,并采用遗传算法对模型进行求解,以从候选资源集合中为任务阶段匹配合适的资源。算例结果表明,该模型能够有效实现N-N的任务-资源集成调配,使得协同物流网络的整体效益达到最优。     

Digital resources inviting changes in mid-adopting teachers’ practices and orchestrations

Digital resources offer opportunities to improve mathematics teaching and learning, but meanwhile may question teachers’ practices. This process of changing teaching practices is challenging for teachers who are not familiar with digital resources. The issue, therefore, is what teaching practices such so-called ‘mid-adopting’ mathematics teachers develop in their teaching with digital resources, and what skills and knowledge they need for this. To address this question, a theoretical framework including notions of instrumental orchestration and the TPACK model for teachers’ technological pedagogical content knowledge underpins the setting-up of a project with twelve mathematics teachers, novice in the field of integrating technology in teaching. Technology-rich teaching resources are provided, as well as support through face-to-face group meetings and virtual communication. Data include lesson observations and questionnaires. The results include a taxonomy of orchestrations, an inventory of skills and knowledge needed, and an overview of the relationships between them. During the project, teachers do change their orchestrations and acquire skills. On a theoretical level, the articulation of the instrumental orchestration model and the TPACK model seems promising.     

Humans-with-media and continuing education for mathematics teachers in online environments

This paper begins by situating online mathematics education in Brazil within the context of research on digital technology over the past 25?years. I argue that Brazilian research on technology in mathematics education can be divided into four phases, and then present an example that ??blends?? aspects of the second and third phases. Phase two can be characterized by research with software designed to address traditional mathematics topics, such as functions, while the third phase is characterized by online courses. The data presented show creative solutions for a problem designed for collectives of humans-with-function-software. The paper is analyzed from a perspective that emphasizes the role of different technologies as teachers and professors collaborate to produce knowledge about the use of mathematical software in regular face-to-face classrooms. A model of online education is presented. Finally, the paper discusses how technology may change collaboration and teaching approaches in continuing education, as it allows for greater integration of online learning with teachers?? classroom activities in schools. In this case, the online platform plays an active role in the learning collective composed of humans-with-media.     

Online paths in mathematics teacher training: new resources and new skills for teacher educators

This paper considers the work carried out by online teacher educators and their professional development. We use the theoretical perspective of the documentational approach that focuses, in this case, on the interaction between teacher educators and the resources they use for their online training work. We thus study the following issues: (1) What kinds of resources do online teacher educators need, and how are such resources modified according to the educators?? specific skills and needs? (2) What specific skills are needed for setting up online training for mathematics teachers and how do these skills evolve as teacher education resources are used? We consider both questions simultaneously, while presenting results from a study within a specific teacher training programme in France that proposes ??training paths?? on a national platform. These ??paths?? are resources designed for teacher educators. We follow the appropriation of two training paths by two educator teams. The ways in which these educator teams were able to appropriate the paths give insights into the teacher educators?? skills and, as well, into the resources they need. By looking at their use of resources (as online mathematics teacher educators), we observe and analyse professional geneses, leading to the development of new skills.     

CONJECTURING IN OPEN GEOMETRIC SITUATIONS USING DYNAMIC GEOMETRY: AN EXPLORATORY CLASSROOM EXPERIMENT

In this paper I describe a classroom teaching experiment carried out with a class of Year 10 students. This experiment was twofold. On the one hand it was aimed at developing and trying out a new mode of working in the classroom, taking into account the possibilities offered by dynamic geometry software as support in the conjecturing and proving process in geometry. On the other hand, from the research point of view, it provided the possibility of testing out in the classroom a theoretical model describing and interpreting students’ use of Cabri-Géomètre in open geometric problems, with a particular focus on dragging. The main findings relate to the evolution in the use of dragging in Cabri and the production of rich conjectures, which can provide the basis for development and evolution towards the proving process.     

A multicriteria decision model for assigning priority classes to activities in project management

Problems related to the management of construction projects are addressed in many studies. Falling behind schedule and being over budget are examples of bad results, due to the uncertainties and the dynamic environment of the construction process. This study proposes a decision model for helping project managers to focus on the main tasks of a project network during the life cycle of a project based on a MCDA (multiple criteria decision analysis) method. The model assigns priorities classes to activities in project management, taking into account several points of view. The model is based on the ELECTRE TRI-C method, which permits activities to be assigned to categories. As the environment is very dynamic, the model was built taking into consideration changes that may occur while a project is being carried out, and therefore the model must be reassessed during the project life cycle. Furthermore, the model supports a decision making environment where responsibilities are distributed amongst project team members and it brings the benefit of developing disciplines that lead to the team??s greater effectiveness. An application of the model, based on a realistic situation, is presented in the context of a construction project in order to demonstrate the use of the model. The results show that by using the model, managers can improve their performance with regard to controlling project activities.