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In this paper, we consider theories about processes of visual perception and perception-based knowledge representation (VPR) in order to explain difficulties encountered in figural processing in junior high school geometry tasks. In order to analyze such difficulties, we take advantage of the following perspectives of VPR: (1) Perceptual organization: Gestalt principles, (2) recognition: bottom-up and top-down processing; and (3) representation of perception-based knowledge: verbal vs. pictorial representation, mental images and hierarchical structure of images. Examples given in the paper were mostly taken from Gal's study () which aimed at identifying and analyzing Problematic Learning Situations (after Gal & Linchevski, ) in junior high school geometry classes. Gal's study () suggests that while this theoretical perspective became part of teachers' pedagogic content knowledge, the teachers were aware of their students' thinking processes and their ability to analyze and cope with their students’ difficulties in geometry was improved. [ABSTRACT FROM AUTHOR]

Our research is concerned with teacher’s knowledge, and especially with teacher’s processes of learning, in the classroom, from observing and interacting with students’ work. In the first part of the paper, we outline the theoretical framework of our study and distinguish it from some other perspectives. We argue for the importance of distinguishing a kind of teacher’s knowledge, which we call didactic knowledge. In this paper, we concentrate on a subcategory of this knowledge, namely observational didactic knowledge, which grows from teacher’s observation and reflection upon students’ mathematical activity in the classroom. In modeling the processes of evolution of this particular knowledge in teachers, we are inspired, among others, by some general aspects of the theory of didactic situations. In the second part of the paper, the model is applied in two case studies of teachers conducting ordinary lessons. In conclusion, we will discuss what seems to be taken into account by teachers as they observe students’ activity, and how in-service teacher training can play a role in modifying their knowledge about students’ ways of dealing with mathematical problems. [ABSTRACT FROM AUTHOR]

This paper describes a task-based dynamic geometry platform that is able to record student responses in a collective fashion to pre-designed dragging tasks. The platform provides a new type of data and opens up a quantitative dimension to interpret students' geometrical perception in dynamic geometry environments. The platform is capable of generating a collective image map of student geometrical perceptions for a pre-designed dragging task. This map is interpreted as students' qualitatively different ways of perceiving a geometrical phenomenon under the drag mode, ways which are quantified and categorized in a collective way. The idea of task perceptual landscape is proposed to facilitate discussion on the pedagogical significance of this platform. Specifically, a task case is presented and analysed in which a methodology is developed that provides a way to classify students' geometrical perceptions with respect to the task. The task perceptual landscape is interpreted as a collective example space of student perception of a task. Furthermore, an idea of personal example space is developed through the findings from a qualitative study for the same task. This brings about discussion on possible pedagogical correlation between the quantitative and qualitative aspects of the platform. [ABSTRACT FROM AUTHOR]

Although proof comprehension is fundamental in advanced undergraduate mathematics courses, there has been limited research on what it means to understand a mathematical proof at this level and how such understanding can be assessed. In this paper, we address these issues by presenting a multidimensional model for assessing proof comprehension in undergraduate mathematics. Building on Yang and Lin's (Educational Studies in Mathematics 67:59-76, ) model of reading comprehension of proofs in high school geometry, we contend that in undergraduate mathematics a proof is not only understood in terms of the meaning, logical status, and logical chaining of its statements but also in terms of the proof's high-level ideas, its main components or modules, the methods it employs, and how it relates to specific examples. We illustrate how each of these types of understanding can be assessed in the context of a proof in number theory. [ABSTRACT FROM AUTHOR]

The geometry textbooks of the new system of secondary education in Iran differed dramatically from the old ones considering the aims, the visions, the content, the approach, and the educational purposes. Four hundred eighty mathematics teachers participated in a nationwide professional development program conducted by the author to facilitate the implementation of the new changes. Based on this program, a research study was designed to investigate the teachers’ beliefs about a new reform in high school geometry as a result of the professional development program. The data of the study were collected through teachers’ reflective writings, project works, group discussions, reflective questionnaires, and oral communications. The purpose of this paper is to draw upon this study and to document mathematics teachers’ beliefs about new changes. [ABSTRACT FROM AUTHOR]

Proof and reasoning are fundamental aspects of mathematics. Yet, how to help students develop the skills they need to engage in this type of higher-order thinking remains elusive. In order to contribute to the dialogue on this subject, we share results from a classroom-based interpretive study of teaching and learning proof in geometry. The goal of this research was to identify factors that may be related to the development of proof understanding. In this paper, we identify and interpret students' actions, teacher's actions, and social aspects that are evident in a classroom in which students discuss mathematical conjectures, justification processes and student-generated proofs. We conclude that pedagogical choices made by the teacher, as manifested in the teacher's actions, are key to the type of classroom environment that is established and, hence, to students' opportunities to hone their proof and reasoning skills. More specifically, the teacher's choice to pose open-ended tasks (tasks which are not limited to one specific solution or solution strategy), engage in dialogue that places responsibility for reasoning on the students, analyze student arguments, and coach students as they reason, creates an environment in which participating students make conjectures, provide justifications, and build chains of reasoning. In this environment, students who actively participate in the classroom discourse are supported as they engage in proof development activities. By examining connections between teacher and student actions within a social context, we offer a first step in linking teachers' practice to students' understanding of proof. [ABSTRACT FROM AUTHOR]