Formal Mathematics for Mathematicians

The collection of works for this special issue was inspired by the presentations given at the 2011 AMS Special Session on Formal Mathematics for Mathematicians: Developing Large Repositories of Advanced Mathematics. The issue features a collection of articles by practitioners of formalizing proofs who share a deep interest in making computerized mathematics widely available.     

Mathematical Challenge in the Eyes of the Beholder: Mathematics Teachers’ Views

Abstract

This study is based on our belief that mathematics should be challenging in any classroom and that mathematical challenge is among the central factors that determine the quality of mathematics lessons. Choosing challenging mathematical problem for the students is central in teachers’ work while their conception of mathematical challenge can determine the quality of a mathematics lesson. At the same time, little is known about teachers’ views on mathematical challenge. Thus, we explored conceptions of mathematical challenge in two groups of experienced mathematics teachers. The first group (N₁ = 9) was asked to define the notion of mathematical challenge and give examples of challenging mathematical tasks. Later the members of the group discussed these examples and definitions. A written response questionnaire was administered to a second group of teachers (N₂ = 41) based on answers given by teachers in the first group. We found that the teachers have a broad conception of mathematical challenge and appreciate the relativity of mathematical challenge but are not always convinced that it is possible to incorporate challenging mathematics in everyday teaching in the classroom.     

“I Fall Asleep in Class … But Physics Is Fascinating”: The Use of Large-Scale Longitudinal Data to Explore the Educational Experiences of Aspiring Girls in Mathematics and Physics

This article explores how students' aspirations to study mathematics or physics in post-16 education are associated with their perceptions of their education, their motivations, and the support they feel they received. The analysis is based on the responses of around 10,000 students in England in Year 8 (age 12–13) and then in Year 10 (age 14–15). The students were first surveyed during 2008–2009 and then followed up in 2010–2011. t-tests revealed a decline in their perceptions of their mathematics and physics education. Factor analyses indicated subject-specific constructs that were associated with gender aspiration groups (i.e., high-aspiring girls, high-aspiring boys, low-aspiring girls, low-aspiring boys). High-aspiring girls were more likely than low-aspiring boys to be positive about mathematics/physics education, motivation in these subjects, and support received. However, high-aspiring girls were less likely than high-aspiring boys to be encouraged by their teachers and families to continue with these subjects post-16 and had lower self-concepts, intrinsic valuations, and perceptions of lessons. Low-aspiring girls reported the least favorable views of their mathematics/physics education of all four gender aspiration groups. Findings were generally similar for mathematics and physics, although students overall responded more favorably to mathematics than to physics. The quantitative findings are illustrated with extracts from longitudinal interviews (ages 15, 16, and 17) of two high-aspiring girls.     

Provision of awareness of learners’ emotions through visualizations in a computer interaction-based environment

One of the challenges of intelligent systems for education is to use low-level data collected in computer environments in the form of events or interactions to infer information with high-level significance using artificial intelligence techniques, and present it through visualizations in a meaningful and effective way. Among this information, emotional data is gaining track in by instructors in their educational activities. Many benefits can be obtained if an intelligent systems can bring teachers with knowledge about their learner’s emotions, learning causes, and learning relationships with emotions. In this paper, we propose and justify a set of visualizations for an intelligent system to provide awareness about the emotions of the learners to the instructor based on the learners’ interactions in their computers. We apply these learner’s affective visualizations in a programming course at University level with more than 300 students, and analyze and interpret the student’s emotional results in connection with the learning process.     

基于学科应用的离散数学课程教学方法探讨

针对传统授课方式中存在的问题,从教学模式、课程导入方式、教学内容等方面对离散数学课程的教学改革进行了探讨。改革的目标是强化学生对离散数学课程在计算机学科中应用的了解,加深学生对离散数学的重要性的理解,使学生的前后知识体系得到衔接,从而提高学生的学习兴趣。最后给出了离散数学在后续课程中的一些应用实例。     

The design and application of technology-based courses in the mathematics classroom

It is difficult to examine the effectiveness of technology-based courses (TBC) without understanding the design and application in classrooms. There is evidence of disconnects among the theory for designing, the theory used to apply TBCs in classrooms, and the theory used to research and evaluate TBCs [Hickey, D. T. (1997). Motivation and contemporary socio-constructivist instructional perspectives. Educational Psychologist, 32(3), 175–193]. Comments provided by administrators, teachers and students may lead researchers to determine whether or not the original TBC course goals have been attained. In this paper, we first discuss examples of theoretical disconnects found in other technology-based research [Aleven, V. E., Stahl, E., Schworm, S., Fischer, F., & Wallace, R. (2003). Help seeking and help design in interactive learning environments. Review of Educational Research, 73(3), 277–320; Hickey, D. T., & McCaslin, M. (2001). A comparative, sociocultural analysis of context and motivation. In S. Volet, & S. Järvelä (Eds.), Motivation in learning contexts: Theoretical advances and methodological implications (pp. 33–55). Elmsford, NY: Pergamon Press]. Then, we describe the course design of the mathematics TBC used in this study, the reasons the principal and the teacher’s believe the software will benefit their mathematics students, and the perceptions of mathematics students regarding the effectiveness of the technology in their classroom. In conclusion, we will discuss how this preliminary qualitative data shaped our future research questions.     

Bringing down the master's house with the master's tools

Abstract

Examples, illustrations, occurrences, and instances play a central role in the learning of mathematics. Deliberately getting students to generate examples is a particularly powerful tool in teaching, but, while some of the possibilities they offer are used by many teachers, their full potential is rarely exploited. The authors came to this notion through reflection on their experience of observing teachers in classrooms, of getting students to take initiatives, and of creating examples while working on their own mathematics. They examine the roles played by examples constructed and generated by students, give illustrations and analyses of the use of this tool, and develop a theory for the act of exemplification as an act of cognition. A framework is developed for researchers and practitioners to further this work.     

Social Representations of High School Students About Mathematics Assessment

The perceptions of students about assessment in mathematics classes have been sparsely investigated. In order to fill this gap, this qualitative study aims to identify the social representations (understood as the system of values, ideas, and practices about a social object) of high school students regarding assessment in mathematics. We used semistructured focus group interviews to obtain data (N = 50). The data were examined using theoretical thematic analysis with the assistance of the specialized ATLAS.ti software. Eight themes or social representations emerged (e.g., assessment that measures the students' learning or assessment that measures the advances or what has been acquired). The results are consistent with literature that shows that students' representations about assessment in mathematics are closely linked to their representations of mathematics itself and the learning of it and also to their representation of what is fair and what is not.     

A Theorem and Its Different Proofs: History,Mathematics Education,and the Semiotic-Cultural Perspective

Abstract

Several studies claim that integrating the history of mathematics into mathematics education is an important matter. In this article, some theoretical approaches are presented and the importance of the correct cultural contextualization is emphasized. Our purpose is to discuss, from an educational viewpoint, the problem of knowledge development within specific cultural contexts; cultural aspects play a cognitive and epistemological role in the way we think mathematically. More specifically, we provide a comparison of some strategies used by mathematicians in different periods in order to prove that primes are infinitely many. We conclude that historical examples should be understood in their cultural and social context and that the standards of symbolization and rigor depend on this context. In order to emphasize the use of history in education as a real tool for the teacher, we propose an outline of educational opportunities. The perspective presented requires a good level of epistemological skill on the part of teachers; therefore, it should be considered in teacher education.     

Encouraging Citizenship in Science Education: Continuing Questions and Hopeful Possibilities

Abstract

This special issue of the Canadian Journal of Science, Mathematics and Technology Education invokes questions intended to further the discourse of citizenship in science and mathematics education, such as, How do we define citizen and democracy? Is our call for student action hypocritical? Does positioning school science through the work of Rancière present a “straw man” argument for change? To what extent does the ghost of John Dewey animate and inform a “wild pedagogy” in science education? Challenging the view of the science and mathematics curriculum as a barrier to overcome, this article argues that possibilities for developing citizenship and critical thinking can be found and developed in existing curriculum formations and practices of school science and mathematics education.     

Rethinking Technology-Enhanced Physics Teacher Education: From Theory to Practice

This article discusses how modern technology, such as electronic response systems, PeerWise system, data collection and analysis tools, computer simulations, and modeling software can be used in physics methods courses to promote teacher-candidates' professional competencies and their positive attitudes about mathematics and science education. We show how modeling technology-enhanced deliberate pedagogical thinking in physics methods courses can improve teacher-candidates' subject-specific pedagogical knowledge and their positive attitudes about science learning. We also discuss potential challenges that must be addressed in order to help teacher-candidates successfully implement these pedagogies during the practicum and in their early years of teaching.     

Lyra: An Interactive Visualization Design Environment

We present Lyra, an interactive environment for designing customized visualizations without writing code. Using drag‐and‐drop interactions, designers can bind data to the properties of graphical marks to author expressive visualization designs. Marks can be moved, rotated and resized using handles; relatively positioned using connectors; and parameterized by data fields using property drop zones. Lyra also provides a data pipeline interface for iterative, visual specification of data transformations and layout algorithms. Visualizations created with Lyra are represented as specifications in Vega, a declarative visualization grammar that enables sharing and reuse. We evaluate Lyra's expressivity and accessibility through diverse examples and studies with journalists and visualization designers. We find that Lyra enables users to rapidly develop customized visualizations, covering a design space comparable to existing programming‐based tools.     

Learning About Ourselves: A Review of The Mathematics Teacher in the Digital Era

ABSTRACT

Mathematics Education in the Digital Era is a title that implies the ubiquity of digital technology in our everyday lives. In fact, it can be argued that digital technology is already ubiquitous for many stakeholders in mathematics education. Volume 2 of this series is titled The Mathematics Teacher in the Digital Era, which implies an exploration of the mathematics teacher as individual agent or actor in the experience of teaching and learning mathematics. What is exciting is the book editors' potentially unintended insights into teachers' affective sense, their beliefs, and their subsequent decisions for future pedagogical effort.     

The teacher's responsibility in whole‐class debriefing of mathematical activity

Abstract

Prompted by recent moves in the United Kingdom to guide teachers’ practices in whole‐class, direct interactive teaching, in this article, we offer an opportunity for North American mathematics educators to reflect on possibilities for whole‐class teaching of mathematics. We focus particularly on the plenary aspect of lessons—what might be considered the debriefing of mathematical activity—and specifically on the teacher's responsibility during those sessions, both to his or her students and to the authenticity of the discipline of mathematics. Drawing on data from a Grade‐3 classroom and invoking complexity science as a theoretical lens to explore the classroom as a complex learning system, we present implications for teaching in whole‐class debriefings of mathematical activity.     

Learning with dynamic and static visualizations: Realistic details only benefit learners with high visuospatial abilities

Learning environments can nowadays easily be enriched with different presentation formats of visualizations, because computer graphics technology is constantly and rapidly developing. This study investigates the effectiveness of dynamic compared to static visualizations. Moreover, the influence of realistic details in the visualizations as well as learners’ prerequisites in terms of their visuospatial abilities were addressed. Eighty university students were randomly assigned to four conditions of a two-by-two between subjects design with the two independent variables dynamism and realism. Learning outcomes were measured by means of a verbal factual knowledge test about the terminology and visuospatial details and a pictorial recognition test about the dynamic processes. Data analyses revealed no effects for factual knowledge. With respect to recognition, learners with dynamic visualizations outperformed learners with static visualizations. Furthermore, there was an interaction between learners’ visuospatial abilities and the degree of realism in the visualization: learners with lower visuospatial abilities showed better recognition with schematized visualizations, whereas learners with higher visuospatial abilities showed better recognition with realistic visualizations. The results imply that when designing instructional materials, both the type of knowledge that has to be acquired as well as learners’ prerequisites such as their visuospatial abilities need to be considered.     

Where is the teacher? Digital analytics for classroom proxemics

The term Classroom Proxemics refers to how teachers and students use classroom space, and the impact of this and the spatial design on learning and teaching. This study addresses the divide between, on the one hand, substantial work on proxemics based on classroom observations and, on the other hand, emerging work to design automated feedback that helps teachers identify salient patterns in their use of the classroom space. This study documents how digital analytics were designed in service of a senior teacher's practice-based inquiry into classroom proxemics. Indoor positioning data from four teachers were analysed, visualized and used as evidence to compare three distinct learning designs enacted in a physics classroom. This study demonstrates how teachers can make effective use of such visualizations, to gain insight into their classroom practice. This is evidenced by (a) documenting teachers' reflections on visualizations of positioning data, both their own and that of peers and (b) identifying the types of indicator (operationalized as analytical metrics) that foreground the most useful information for teachers to gain insight into their practice.     

适用于多模式虚拟学习的交互医学可视化及模拟

医学图像及其衍生的知识是医学教育中的一个关键学习部分，透过不同的图像模式，町以使医生及医科学生更有效地学习人体的体内结构．而计算机可视化技术最近的发展正满足了医生、外科医生及医科学生与日俱增的需求，为深入地了解人体生理学及病理学提供了宝贵的资料．本文提出了一个网上交互医学学习系统，并提供了一个虚拟现实工具包以支持二维医学图像序列观察、体数据和表面数据的三维可视化、目视放大镜．以及医学数据的交互变形模拟．其中人体组织物理特性的模拟以有限元法为基准．通过优化而获得有关的变形模型参数．该工具包可应用于个性化的二维及三维数据场可视化、优化的学习及手术规划和在线的交互医学教育．     

Mathematica软件在数学教学中的应用探索

Mathematica软件是专门进行数学类计算的软件,其主要功能是符号运算功能和绘图功能.在高等数学的教学过程中引入Mathematica软件,有利于增强教学内容的直观性,激发学生学习数学的兴趣,提高解决实际问题的能力.     

Computer-based instruction and generative strategies: Conceptual framework & illustrative example

Adult students who are enrolled in higher education must experience computer-based instruction that is well-designed in terms of both efficiency and relevance. Published practical examples of processes that will result in such instruction are rare. This article begins by describing the needs of adult students who are enrolled in higher education. Then, this article describes a project in which Morrison, Ross, and Kemp’s (2004) curvilinear instructional design model was used to create computer-based instruction in the field of developmental mathematics. Both the design and development phases are described. The article concludes with implications for others who might use a similar approach in higher education.     

Explanatory and illustrative visualization of special and general relativity

This paper describes methods for explanatory and illustrative visualizations used to communicate aspects of Einstein's theories of special and general relativity, their geometric structure, and of the related fields of cosmology and astrophysics. Our illustrations target a general audience of laypersons interested in relativity. We discuss visualization strategies, motivated by physics education and the didactics of mathematics, and describe what kind of visualization methods have proven to be useful for different types of media, such as still images in popular science magazines, film contributions to TV shows, oral presentations, or interactive museum installations. Our primary approach is to adopt an egocentric point of view: the recipients of a visualization participate in a visually enriched thought experiment that allows them to experience or explore a relativistic scenario. In addition, we often combine egocentric visualizations with more abstract illustrations based on an outside view in order to provide several presentations of the same phenomenon. Although our visualization tools often build upon existing methods and implementations, the underlying techniques have been improved by several novel technical contributions like image-based special relativistic rendering on GPUs, special relativistic 4D ray tracing for accelerating scene objects, an extension of general relativistic ray tracing to manifolds described by multiple charts, GPU-based interactive visualization of gravitational light deflection, as well as planetary terrain rendering. The usefulness and effectiveness of our visualizations are demonstrated by reporting on experiences with, and feedback from, recipients of visualizations and collaborators.