Students’ ratings of teacher practices

In this paper, we explore a novel approach for assessing the impact of a professional development programme on classroom practice of in-service middle school mathematics teachers. The particular focus of this study is the assessment of the impact on teachers’ employment of strategies used in the classroom to foster the mathematical habits of mind and mathematical self-efficacy of their students. We describe the creation and testing of a student survey designed to assess teacher classroom practice based primarily on students’ ratings of teacher practices.     

Measuring Sixth‐Grade Students' Problem Solving: Validating an Instrument Addressing the Mathematics Common Core

This article describes the development of a problem‐solving instrument intended for classroom use that addresses the Common Core State Standards for Mathematics. In this study, 137 students completed the assessment, and their responses were analyzed. Evidence for validity was collected and examined using the current standards for educational and psychological testing. Instrument validation findings regarding internal consistency reliability were high, and multiple forms of validity (i.e., content, response processes, internal structure, relationship to other variables, and consequences of testing) were found to be appropriate. The resulting instrument provides teachers and researchers with a sound tool to gather data about sixth‐grade students' problem solving in the Common Core era.     

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.     

A teacher education experiment to challenge conceptions and practices

This study describes a teacher education experience with grade 5–6 teachers, based on a calculator module within a national program for mathematics in-service teacher education. The aim was to challenge the teachers’ conceptions about the role of the calculator in mathematics teaching and to promote their reflection about professional practices. The research methodology was qualitative and interpretive, with data collection through interviews and observation of teacher education and classroom supervision sessions, as well as analysis of teachers’ portfolios. The results indicate that some teachers are clearly against the use of the calculator in the mathematics classroom, while others allow students to use it in a passive way and some others are very affirmative about its use. The teachers who argue against the use of the calculator seem to predominate, suggesting a great distance between the curriculum orientations and classroom practice. The methodology of the course, combining collective sessions and individual classroom supervision, proved to be fruitful, providing new information, practice and discussion that allowed teachers to analyze different kinds of tasks in which the calculator might be useful, experiment using them in the classroom and reflect about the students’ work. The no imposing and questioning approach used in collective discussions encouraged teachers to assume their own positions; sharing and discussing in the collective reflections during the course stimulated a deeper reflection of their practice. Therefore, in this course, in-service teacher education focused on practice contributed to teachers to reflect on their conceptions and practices.     

Representing context in video records of practice for urban mathematics teacher education

Mathematics education research has not sufficiently theorized about mathematics teacher knowledge and practice, teacher learning, and teacher education in ways that are reflective of the specificities of the sociopolitical contexts of schooling. In the USA, this is particularly important for urban mathematics education. This paper examines the affordances and challenges of representing context in video records of practice, particularly in the urban context, for use in the preparation of mathematics teachers for urban settings. The discussion, grounded in current research and theory relevant to representations of teaching, urban education, and mathematics teacher education, takes up three key issues: how is a focus on the urban context relevant to the design of video records of practice for mathematics teacher education? How can video records support prospective teachers’ understandings of the sociopolitical contexts of mathematics teaching? How does a focus on the urban context impact the meaning teachers make of video records?     

问题驱动 多元表征的概念教学探究——对偶线性规划教学案例设计

数学与应用数学(师范)专业中的《运筹学》具有跨学科、实践性的课程特点,目标在于培养职前教师用数学方法解决实际问题的能力.结合义务教育阶段新课程标准中"四基"的提出这一背景,本文将以线性规划部分(运筹数学)对偶线性规划概念的引入这一知识模块为例,探讨通过问题串形式进行问题驱动、多元表征的概念教学过程.即遵循问题驱动—兴趣驱动—问题意识发展—提出和解决新问题,依据数学与外部联系、数学内部联系两条主线设计教学和学习,探索如何通过问题驱动、多元表征的结构化教学过程引导学生的学习方式发生改变,增强探究学习的动机,发展问题解决能力.课堂教学实践证明效果优于以往单一的讲授式教学法,一定程度上提高了学生的学业成绩、应用问题的兴趣和问题解决意识.     

An “inverse” relationship between mathematics identities and classroom practices among early career elementary teachers: The impact of accountability

This qualitative case study guided by portraiture examines the relationships between three early career elementary teachers’ beliefs about themselves in relation to mathematics (mathematics identities) and their classroom practices. Through autobiographical inquiry, reflective practice, classroom observations, interviews, and artifacts, findings show that all three second grade teachers appeared to have an “inverse” relationship between their mathematics identities and their classroom practices. In this relationship, as negative as they felt about themselves with regards to mathematics, they expended that much more effort to ensure that their students would have positive experiences with it and not be stigmatized by it as they had been. Accountability to schools, students, and parents, to increase student achievement appeared to play an important role in this relationship. Implications for preservice teacher education, inservice professional development, and research on beliefs and practices are discussed.     

Teachers and didacticians: key stakeholders in the processes of developing mathematics teaching

This paper sets the scene for a special issue of ZDM—The International Journal on Mathematics Education—by tracing key elements of the fields of teacher and didactician/teacher-educator learning related to the development of opportunities for learners of mathematics in classrooms. It starts from the perspective that joint activity of these two groups (teachers and didacticians), in creation of classroom mathematics, leads to learning for both. We trace development through key areas of research, looking at forms of knowledge of teachers and didacticians in mathematics; ways in which teachers or didacticians in mathematics develop their professional knowledge and skill; and the use of theoretical perspectives relating to studying these areas of development. Reflective practice emerges as a principal goal for effective development and is linked to teachers’ and didacticians’ engagement with inquiry and research. While neither reflection nor inquiry are developmental panaceas, we see collaborative critical inquiry between teachers and didacticians emerging as a significant force for teaching development. We include a summary of the papers of the special issue which offer a state of the art perspective on developmental practice.     

Journey toward Teaching Mathematics through Problem Solving

Teaching mathematics through problem solving is a challenge for teachers who learned mathematics by doing exercises. How do teachers develop their own problem solving abilities as well as their abilities to teach mathematics through problem solving? A group of teachers began the journey of learning to teach through problem solving while taking a Teaching Elementary School Mathematics graduate course. This course was designed to engage teachers in problem solving during class meetings and required them to do problem solving action research in their classrooms. Although challenged by the course problem solving work, teachers became more comfortable with the mathematics and recognized the importance of group work while problem solving. As they worked with their students, teachers were more confident in their students' abilities to be successful problem solvers. For some teachers, a strong problem solving foundation was established. For others, the foundation was more tentative.     

Integrating Science,Mathematics, and Technology in Middle School Technology‐Rich Environments: A Study of Implementation and Change

The GTECH project, funded through a grant from the GTE Foundation, prepared school teams of science, mathematics and technology teachers and an administrator to set goals for their local schools regarding implementation of electronic technology and integration of content across curricular areas. A variety of teacher‐centered staff development strategies were used to enable participants to achieve local school objectives, model and encourage active learning environments involving technology, develop integrated curriculum and provide training to their peers. GTECH staff provided workshops and summer institutes based on teacher feedback and classroom observations. Data from the Stages of Concerns Questionnaire assisted the staff in designing effective staff development activities. Over the 2‐year period, teacher teams developed and implemented integrated instructional materials and developed skills in using HyperStudio, PowerPoint, telecommunications applications, and instructional resources from the Internet. They also linked instruction to new state and national standards in science, mathematics, and technology. GTECH teachers reported that their students have expanded their knowledge and skill in problem solving, teamwork, technical expertise, and creativity.     

Dimensions of the coordinate functions of space-filling curves

The graphs of coordinate functions of space-filling curves such as those described by Peano, Hilbert, Pólya and others, are typical examples of self-affine sets, and their Hausdorff dimensions have been the subject of several articles in the mathematical literature. In the first half of this paper, we describe how the study of dimensions of self-affine sets was motivated, at least in part, by these coordinate functions and their natural generalizations, and review the relevant literature. In the second part, we present new results on the coordinate functions of Pólya's one-parameter family of space-filling curves. We give a lower bound for the Hausdorff dimension of their graphs which is fairly close to the box-counting dimension. Our techniques are largely probabilistic. The fact that the exact dimension remains elusive seems to indicate the need for further work in the area of self-affine sets.     

Middle School Mathematics Teachers’ Perceptions of the Common Core State Standards for Mathematics and Its Impact on the Instructional Environment

This study examines Middle School Mathematics Teachers’ (MSMTs') (N = 1,241) perceptions of the Common Core State Standards for Mathematics (CCSSM) and its impact on the instructional environment. A total of eight factors appear in the data. These factors include professional support, teachers’ use of district‐adopted and non‐district‐adopted curricular resources, influence of CCSSM assessment and teacher evaluation on the instructional environment, influence of the CCSSM on classroom instruction, teachers’ planning practices, and perceived rigor of the CCSSM. The data suggest that teachers’ use of digital resources located online was disconnected from their district‐adopted resources suggesting a lack of curricular coherence. MSMTs note that the CCSSM had caused them to incorporate more reform‐oriented practices such as exploration and productive struggle into their daily instruction. MSMTs also perceive that the CCSSM includes new content that is more rigorous than previous state standards. Sampled MSMTs state that while state assessments will measure CCSSM content, they are less likely to include more complex problems or the standards for mathematical practice. Teachers are more likely to read teacher resources than student textbook activities online and to use digital resources for remediation instead of inquiry activities. Over one‐third of MSMTs wanted more CCSSM professional development.     

Students’ and Teachers’ Conceptual Metaphors for Mathematical Problem Solving

Metaphors are regularly used by mathematics teachers to relate difficult or complex concepts in classrooms. A complex topic of concern in mathematics education, and most STEM‐based education classes, is problem solving. This study identified how students and teachers contextualize mathematical problem solving through their choice of metaphors. Twenty‐two high‐school student and six teacher interviews demonstrated a rich foundation for these shared experiences by identifying the conceptual metaphors. This mixed‐methods approach qualitatively identified conceptual metaphors via interpretive phenomenology and then quantitatively analyzed the frequency and popularity of the metaphors to explore whether a coherent metaphorical system exists with teachers and students. This study identified the existence of a set of metaphors that describe how multiple classrooms of geometry students and teachers make sense of mathematical problem solving. Moreover, this study determined that the most popular metaphors for problem solving were shared by both students and teachers. The existence of a coherent set of metaphors for problem solving creates a discursive space for teachers to converse with students about problem solving concretely. Moreover, the methodology provides a means to address other complex concepts in STEM education fields that revolve around experiential understanding.     

The beliefs of ‘Tomorrow's Teachers’ about mathematics: precipitating change in beliefs as a result of participation in an Initial Teacher Education programme

Mathematics education research has given increasing attention to the role of affective factors in the learning process. While 'affect' is used to refer to a variety of aspects including feelings, emotions, beliefs, attitudes and conceptions, this paper focuses on 'beliefs' of elementary pre-service teachers. In particular, the study evaluates the effect of participation in a reform-based elementary pre-service teacher education (referred to as Initial Teacher Education (ITE)) programme on participants' 'beliefs about the nature of mathematics'. This was completed using two (sub)scales of the Aiken's Revised Mathematics Scale measuring Enjoyment of Mathematics (E) and belief in the Value of Mathematics (V). Both scales were administered before and after participants completed the mathematics education programme, which consisted of 5 compulsory and consecutive modules. This study reveals that entry-level pre-service teachers report generally positive beliefs about the value of and enjoyment in doing mathematics. The findings challenge previous research, which report the tendency of teachers' beliefs to be resistant to change while in teacher education and suggest that it is possible for ITE mathematics education programmes to stimulate improvement in beliefs and attitudes among participants. Particular programme features are identified as instrumental in this positive change in beliefs about mathematics.     

On the differentiability of the coordinate functions of Póyla's space-filling curve

The coordinate functions of Pólya's space-filling curve are nowhere differentiable if the smaller angle of the triangular target set is between 30° and 45°, limits included; are not differentiable almost everywhere and have derivative zero on a non-denumerable set if the angle is between 15° and 30°, lower limit included; and have derivative zero almost everywhere if the angle is less than 15°.With 5 Figures1925–1994     

Influence of a Mathematics Teachers' Circle on Elementary Teachers' Use of Problem Solving

Math teachers' circles are a form of professional development that is recommended by the Conference Board of the Mathematical Sciences in their publication Mathematical Education of Teachers II (2012). However, little research has been published on how effective math teachers' circles are in advancing the mathematical knowledge of teachers and influencing their actual teaching practice. This study reports how 25 elementary teachers from schools with minority and economically disadvantaged populations responded to a math circle experience. A pretest and posttest showed a significant increase in problem‐solving ability. Through analysis of journals, all but 4 of the 25 teachers indicated that participation in the math circle increased their understanding of problem solving and advanced their use of problem solving in the classroom.     

Simple random walk on the line in random environment

Summary We obtain strong limiting bounds for the maximal excursion and for the maximum reached by a random walk in a random environment. Our results derive from a simple proof of Pólya's theorem for the recurrence of the random walk on the line. As applications, we obtain bounds for the number of visits of the random walk at the origin.     

The Relationship Between Teacher and Student Beliefs About Mathematics

In this study the relationship between teacher (n = 7) beliefs about mathematics, the learning and teaching of mathematics and their respective students' beliefs about mathematics (n = 158) are examined. The data were collected by means of two instruments specifically designed to measure belief systems about mathematics. Teacher scores were adjusted so that a higher score reflected beliefs in aligntnent with the National Council of Teachers of Mathematics (NCTM) Standards. Results indicated that the students of teachers whose beliefs were in alignment with the NCTM Standards had significantly different beliefs about factors that lead to success in mathematics than did other students. Specifically these students felt that working hard to solve problems and striving for understanding would lead to success. No student differences were found for subscales of ego orientation, competitiveness, interest and extrinsic factors such as neatness and cooperation. These findings suggest that this group of teachers practiced what they believed and that these practices affected what their students believed about mathematics. We suggest that using these two assessments in tandem give a clearer picture of the mathematical environment within a classroom and can be used in professional development workshops to initiate teacher reflection about classroom practices.     

An analytic approach for the analysis of rotations in fringe-balanced binary search trees

This paper presents an analytic approach to the construction cost of fringe-balanced binary search trees. In [7], Mahmoud used a bottom-up approach and an urn model of Pólya. The present method is top-down and uses differential equations and Hwang's quasi-power theorem to derive the asymptotic normality of the number of rotations needed to construct such afringe balanced search tree. We also obtain the exact expectation and variance with this method. Although Pólya's urn model is no longer needed, we also present an elegant analysis of it based on an operator calculus as in [4].This research was supported by the Austrian Research Society (FWF) under the project number P12599-MAT.