Classroom video in teacher professional development program: community documentational genesis perspective

We examine classroom video recordings as a means of supporting the learning of teacher communities. Drawing on a longitudinal professional development program for middle years mathematics teachers in the USA, we first outline two contrasting episodes in which the teachers analyzed same segments of classroom video in two different points in the program, 2 years apart. We document that the teachers considered dramatically different aspects of video-recorded instruction as relevant to their professional interests and learning in the two episodes. We then analyze the episodes, and the intervening developments, from point of view of the community documentational genesis. In doing so, we highlight the teacher community’s creation of shared repertoire of ways of reasoning.     

Using video to teach future teachers to learn from teaching

Video is commonly used in teacher preparation programs. Teacher educators use video for various purposes. In this study, we describe the Learning to Learn from Mathematics Teaching project. In this project, video is used to develop pre-service teachers’ (PSTs) orientations, knowledge and skills for analyzing and reflecting on mathematics teaching in ways that generate knowledge for improvement. We discuss the ways we have used video in a course aimed at developing elementary PSTs’ abilities to learn from teaching. In addition, we report on a study that investigated PSTs’ changes in lesson analysis abilities as a result of participating in the course.     

Impacts of professional development focused on teaching engineering applications of mathematics and science

With the recent national emphasis on preparing children for future careers in science, technology, engineering, and mathematics, K-12 teachers are being called upon to include engineering in their instruction. This study explores the impacts of a summer professional development (PD) program focused on the engineering applications of mathematics and science on in-service K-12 teachers' (a) personal engineering efficacy, (b) engineering teaching efficacy, and (c) perceived barriers to teaching engineering. This quantitative study revealed that a single engineering-focused PD could increase teachers' personal engineering efficacy and engineering teaching efficacy and reduce particular perceived barriers to teaching engineering. No differences existed in pre- to post workshop assessment scores based on grade level taught, gender, or years of teaching experiences. However, pre- to post workshop assessment differences existed between participants depending on the discipline they taught and wether or not they had previously used engineering activities in their classrooms. These findings suggest that a single engineering PD can have significant impacts on in-service teachers' personal engineering efficacy, engineering teaching efficacy, and perceived barriers to teaching engineering, but a one-size-fits-all approach to such PD is not equally effective for all participants.     

Using comics-based representations of teaching, and technology, to bring practice to teacher education courses

This article situates comic-based representations of teaching in the long history of tensions between theory and practice in teacher education. The article argues that comics can be semiotic resources in learning to teach and suggests how information technologies can support experiences with comics in university mathematics methods courses that (a) help learners see the mathematical work of teaching in lessons they observe, (b) allow candidates to explore tactical decision-making in teaching, and (c) support preservice teachers in rehearsing classroom interactions.     

Using negative curvature in solving nonlinear programs

Minimization methods that search along a curvilinear path composed of a non-ascent negative curvature direction in addition to the direction of steepest descent, dating back to the late 1970s, have been an effective approach to finding a stationary point of a function at which its Hessian is positive semidefinite. For constrained nonlinear programs arising from recent applications, the primary goal is to find a stationary point that satisfies the second-order necessary optimality conditions. Motivated by this, we generalize the approach of using negative curvature directions from unconstrained optimization to equality constrained problems and prove that our proposed negative curvature method is guaranteed to converge to a stationary point satisfying second-order necessary conditions.     

Opening the closed text: the poetics of representations of teaching

I provide an analysis of the uses of representations of teaching by embedding such representations within Umberto Eco??s theory of the open work, in which literary works are understood not as carriers of a string of meanings, controlled by the author (??closed works??), but rather as fields of meaning. I contend that the well-established tradition of using representations of teaching for pedagogical purposes corresponds to the use of closed works. In contrast, their use for research purposes corresponds to the use of open works. I develop these considerations through an analysis of several representations of teaching, and show how features of these representations work variously to either open or close the work. I also provide anecdotal evidence that viewers of such ??open works?? may fall victim to a version of the so-called ??intentional fallacy.?? I discuss the implications of this for the integrity of the research enterprise.     

Using the Knowledge Quartet to develop mathematics content knowledge: the role of reflection on professional development

In this paper I report findings from a four year study of beginning elementary school teachers which investigated development in their mathematical knowledge for teaching (MKT). The study took a developmental research approach, in that the teachers and the researcher collaborated to develop the mathematics teaching of the teachers, while also trying to understand how such development occurred and might be facilitated. The Knowledge Quartet (KQ) framework was used as a tool to support focused reflection on the mathematical content of teaching, with the aim of promoting development in mathematical content knowledge. Although I focused primarily on whether and how focused reflection using the KQ would promote development, it was impossible to separate this from other influences, and in this paper I discuss the ways in which reflection was found to interrelate with other areas of influence. I suggest that by helping the teachers to focus on the content of their mathematics teaching, within the context of their experience in classrooms and of working with others, the KQ framework supported development in the MKT of teachers in the study.     

Transfer of programs from development to runtime environments

The transfer of programs from development systems to efficient run-time environments is generally considered as a compilation process possibly followed by an optimization phase. This order excludes most possibilities to make optimizations on the source language level. This paper describes an alternative transfer method where user interaction is done on a user oriented level in order to support the optimization process. Experiments show that the efficiency gain is considerable.     

Challenges associated with the professional development of didacticians

This commentary paper focuses on challenges associated with the professional development of a particular group of mathematics educators: mathematics educators who work in the field of teaching development with practicing teachers. This group is termed in this themed issue didacticians. The paper offers a conceptual framework for addressing the challenges associated with the professional development of didacticians, and uses the framework to comment on the collection of the papers in this themed issue.     

Designing representations of trigonometry instruction to study the rationality of community college teaching

We describe the process followed to design representations of mathematics teaching in a community college. The end product sought are animated videos to be used in investigating the practical rationality that community college instructors use to justify norms of the didactical contract or possible departures from those norms. We have chosen to work within the trigonometry course, in the context of an instructional situation, ??finding the values of trigonometric functions,?? and specifically on a case of this situation that occurs as instructors and students are working on examples on the board. We describe the design of the material needed to produce the animations: (1) identifying an instructional situation, (2) identifying norms of the contract that are key in that situation, (3) selecting or creating a scenario that illustrates those norms, (4) proposing alternative scenarios that instantiate breaches of those norms, and (5) anticipating justifications or rebuttals for the breaches that could be found in instructors?? reactions. We illustrate the interplay of contextual and theoretical elements as we make decisions and state hypothesis about the situation that will be prototyped.     

Using computer software in the teaching of mechanics

The paper looks at ways of using computer software in the teaching of mechanics. The various reasons for using software are discussed to justify the use of software. A number of examples are then considered to show how different types of software can be used. Examples shown are taken from very specific types of software, more general simulation software and mathematical software. The paper discusses using software to explore mechanics, to challenge ‘misconceptions’, to make links between mathematical representations and motion and to solve non-standard problems. The paper also stresses the need for structured approaches to the use of software.     

Mexican high school students' social representations of mathematics,its teaching and learning

This paper reports a qualitative research that identifies Mexican high school students’ social representations of mathematics. For this purpose, the social representations of ‘mathematics’, ‘learning mathematics’ and ‘teaching mathematics’ were identified in a group of 50 students. Focus group interviews were carried out in order to obtain the data. The constant comparative style was the strategy used for the data analysis because it allowed the categories to emerge from the data. The students’ social representations are: (A) Mathematics is…(1) important for daily life, (2) important for careers and for life, (3) important because it is in everything that surrounds us, (4) a way to solve problems of daily life, (5) calculations and operations with numbers, (6) complex and difficult, (7) exact and (6) a subject that develops thinking skills; (B) To learn mathematics is…(1) to possess knowledge to solve problems, (2) to be able to solve everyday problems, (3) to be able to make calculations and operations, and (4) to think logically to be able to solve problems; and (C) To teach mathematics is…(1) to transmit knowledge, (2) to know to share it, (3) to transmit the reasoning ability, and (4) to show how to solve problems.     

Using algebraic models of programs for detecting metamorphic malwares

Polymorphic and metamorphic viruses are the most sophisticated malicious programs that give a lot of trouble to virus scanners. Each time when these viruses infect new executables or replicate themselves, they completely modify (obfuscate) their signature to avoid being detected. This contrivance poses a serious threat to antivirus software that relies on classical virus-detection techniques: such viruses do not have any stable specific sequence of instructions that one looks for. In the ultimate case, the only characteristic that remains invariable for all generations of the same virus is their functionality (semantics). To all appearance, the only way to detect for sure a metamorphic malicious code is to look for a pattern that has the same semantics as (i.e., equivalent to) some representative sample of the virus. Thus, metamorphic virus detection is closely related to the equivalence-checking problem for programs. In this paper, we outline some new automata-theoretic framework for the designing of virus detectors. Our approach is based on the equivalence-checking techniques in algebraic models of sequential programs. An algebraic model of programs is an abstract model of computation, where programs are viewed as finite automata operating on Kripke structures. Models of this kind make it possible to focus on those properties of program instructions that are widely used in obfuscating transformations. We give a survey (including the latest results) on the complexity of equivalence-checking problem in various algebraic models of programs and estimate thus the resilience of some obfuscating transformation commonly employed by metamorphic viruses.     

Using the Knowledge Quartet to quantify mathematical knowledge in teaching: the development of a protocol for Initial Teacher Education

This study examined trainee teachers' mathematical knowledge in teaching (MKiT) over their final year in a US Initial Teacher Education (ITE) programme. This paper reports on an exploratory methodological approach taken to use the Knowledge Quartet to quantify MKiT through the development of a new protocol to code trainees' teaching of mathematics lessons. This approach extends Rowland's et al. work on the Knowledge Quartet (KQ). Justification for using the KQ to quantify MKiT, and the potential benefits such an attempt might provide those involved with ITE, are discussed. It is suggested that quantified MKiT data based on the Knowledge Quartet can be used to consider MKiT development in novice teachers in order to inform ITE programmes and form new theoretical loops between theory and practice in teacher education.     

Toward professional development for teachers grounded in a theory of decision making

There is now robust evidence that teachers’ and others’ in-the-moment decision making can be modeled and explained as a function of the following: their knowledge and other intellectual, social, and material resources; their goals; and their orientations (their beliefs, values, and preferences). The role of beliefs as they affect teachers’ behavior can thus be described at a level of mechanism—but of necessity in interaction with resources and goals. This paper outlines and exemplifies how resources, goals, and orientations shape teachers’ behavior. It indicates how they are interconnected, and why their evolution is necessarily slow. It then suggests how these understandings can be used as a foundation for mathematics teachers’ professional development, and describes how they are being used to shape a course of participatory professional development for middle school mathematics teachers.     

The impact of integrated STEM professional development on teacher quality

This study of a state‐funded, 3‐year implementation of an integrated STEM professional development (PD) program for teachers from two middle schools in the midwestern U.S. examined if participants in the PD were enabled to transform their practice and perceptions of STEM. An integrated STEM approach includes a focus on the STEM disciplines, along with leveraging social studies/history and English/language arts as important context and tools for solving society’s biggest challenges. Findings in this study indicated that teachers implemented more effective STEM teaching strategies and had more positive perceptions regarding STEM overall. Further, participants became more aware of their personal needs for resources and support to teach through integrated STEM. Implications for research and practice are discussed.