Why and how mathematicians read proofs: an exploratory study

In this paper, we report a study in which nine research mathematicians were interviewed with regard to the goals guiding their reading of published proofs and the type of reasoning they use to reach these goals. Using the data from this study as well as data from a separate study (Weber, Journal for Research in Mathematics Education 39:431–459, 2008) and the philosophical literature on mathematical proof, we identify three general strategies that mathematicians employ when reading proofs: appealing to the authority of other mathematicians who read the proof, line-by-line reading, and modular reading. We argue that non-deductive reasoning plays an important role in each of these three strategies.     

Using manipulatives to support an embodied approach to learning trigonometry in a South African school: A case study

ABSTRACT

Multiple Intelligence Theory suggests that individuals perceive knowledge in eight different ways. This article reports on a study that explored the role of manipulatives in the teaching and learning of trigonometric ratios in grade 10. The approach attempts in addressing three domains of the Multiple Intelligence Theory (linguistic/verbal intelligence, logical/mathematical intelligence and spatial intelligence). The foundation of this research was a case study contained in the interpretative paradigm involving five grade 10 mathematics pupils at a high school in South Africa. The data was collected from: (1) activity sheet containing written responses of pupils; (2) observations; and (3) semi-structured interviews. The data was analysed and it was found that the use of manipulatives in teaching and learning mathematics played a positive role in leaners understanding of trigonometric ratios at grade 10 level. In general the findings of this study supported other research findings that confirm that manipulatives were important mediating tools in the development of conceptual and procedural understanding of mathematical concepts. Besides these pedagogical implications the study proved that the manipulatives effectively consolidated the features of Lesh's model.     

Teaching and learning how to write proofs in Concepts of Geometry

This paper concerns my experience teaching Concepts of Geometry, an inquiry-based course that emphasizes discovery learning, analytical thinking, and individual creativity. The author discusses how to guide the students to recognize the connections among different mathematical ideas; to select the types of reasoning and methods of proof that apply to the problem; to organize their mathematical thinking through communication; to make their own mathematical conjectures; and to analyze and evaluate each other＇s mathematical arguments and proofs. In particular, the comments and the feedback from students are included.     

Learning to support learning together: an experience with the soft systems methodology

An action research approach called soft systems methodology (SSM) was used to foster organisational learning in a school regarding the role of the learning support department within the school and its relation with the normal teaching‐learning activities. From an initial situation of lack of coordination as well as mutual misunderstanding and distrust among some of the actors in the situation, eight months of work yielded as a result a better working environment among the participants, the creation of a volunteer discussion group functioning on a permanent basis, and a clarification and modification of some of the learning support activities. This change can also be described as learning by learning support staff, school teachers and administrators on how to work together, as well as on how to support children's learning in a better way. Nevertheless, another event that occurred after the intervention shows the importance of considering a wider system than that limited by the participants in the intervention.     

Motivational support from digital game-based learning environments (DGBLEs) for scientific topics designed by novice end users

The integration of digital game-based learning environment (DGBLE) enhanced by technology in recent years has been hampered by its complexity and difficulty in realistically addressing teachers’ instructional needs. The pedagogical gap between what teachers think a DGBLE can deliver and what DGBLE can actually offer remains substantial. To preliminarily address the gap, this present qualitative study adopted an end-user-development lens to understand how K-12 teachers as novice end users interpret and design game features in their DGBLEs. The design ideas, situated in teaching scientific subjects, were collected from 19 pre-service teachers in a public university in the United States by semi-structured interviews based on game features identified in the literature. The data analysis, grounded in the ARCS (attention, relevance, confidence, satisfaction) motivational design model, employed the direct content analysis method and revealed the distribution of motivational support among participants’ DGBLEs design ideas. The finding suggested that participants were more likely to incorporate intrinsic motivational support than extrinsic support in their design. This trend could compromise the motivational support of DGBLEs when students need additional extrinsic motives toward the end of learning processes. Further, the study illuminates the importance of incorporating end users’ perspectives early on in the DGBLE design process.     

Open educational resources in support of science learning: tools for inquiry and observation

This article focuses on the potential of free tools, particularly inquiry tools for influencing participation in twenty-first-century learning in science, as well as influencing the development of communities around tools. Two examples are presented: one on the development of an open source tool for structured inquiry learning that can bridge the formal/informal spaces for inquiry learning. This is contrasted with an example of the use of free tools and community development for observation of scientific phenomena supported by open educational resources (OER) with a citizen science perspective. The article provides an assessment of how the availability of the resources has a potential for shaping the communities using OER for science learning and a discussion of the means of supporting inquiry.     

Developing online learning resources: Big data,social networks,and cloud computing to support pervasive knowledge

Utilizing online learning resources (OLR) from multi channels in learning activities promise extended benefits from traditional based learning-centred to a collaborative based learning-centred that emphasises pervasive learning anywhere and anytime. While compiling big data, cloud computing, and semantic web into OLR offer a broader spectrum of pervasive knowledge acquisition to enrich users’ experience in learning. In conventional learning practices, a student is perceived as a recipient of information and knowledge. However, nowadays students are empowered to involve in learning processes that play an active role in creating, extracting, and improving OLR collaborative learning platform and knowledge sharing as well as distributing. Researchers have employed contents analysis for reviewing literatures in peer-reviewed journals and interviews with the teachers who utilize OLR. In fact, researchers propose pervasive knowledge can address the need of integrating technologies like cloud computing, big data, Web 2.0, and Semantic Web. Pervasive knowledge redefines value added, variety, volume, and velocity of OLR, which is flexible in terms of resources adoption, knowledge acquisition, and technological implementation.     

An investigation of SOAR study strategies for learning from multiple online resources

This mixed-methods study investigated the effects of the SOAR study strategy for learning from multiple online resources. SOAR includes the components of selection, organization, association, and regulation. Past research confirmed that college students who study provided or partially provided SOAR materials achieved more than those using their preferred study methods when learning from a single printed (Jairam & Kiewra, 2009) or computer-based (Jairam & Kiewra, 2010) text. The present study was the first to investigate SOAR when college students create their own study materials and must learn from multiple online resources. In the present study, one hundred and thirty-four college students were assigned randomly to the preferred strategy control group or the SOAR strategy experimental group. Following a pre-survey and online training in their respective study strategy, both groups were directed to use their trained strategy to study scientific material available on multiple websites. Following the study period, participants were tested on the online material and then completed a post-survey. Qualitative analyses of pre-survey responses (before training) indicated that participants from both groups commonly use note taking, summarization, and memorization strategies to study online resources. Qualitative analyses of post-survey responses (after training) indicated that the preferred strategies control group used these same strategies to study the experimental material, whereas the SOAR-trained group used SOAR strategies. These qualitative data fit with quantitative data showing that (a) both groups created study materials reflective of their strategy training (preferred or SOAR); (b) SOAR strategy studiers achieved more than preferred strategy studiers on fact, relationship, and concept achievement items; and (c) SOAR-trained studiers had more positive attitudes about their trained methods than preferred-trained studiers.     

学习契约:远程学习效果评价的书面协议

过程性评价是远程学习评价的重要部分.作为过程性评价方式的典型代表,学习契约正越来越受到教学评价理论和实践的关注.为了充分理解和有效操作远程学习中学习契约这一评价方法,本文在整合国内外相关资料的基础上,论述了学习契约的定义与类型,分析了学习契约在远程学习中的特征与优势、基本原则和设计方法,并且提供了四个设计模板.     

Everyday teaching and outdoor learning: developing an integrated approach to support school-based provision

In the UK, teachers are increasingly being encouraged to support and undertake outdoor learning. Despite such support there is a paucity of research that has considered how outdoor learning can be implemented and offered on a regular basis by teachers. The lack of empirical research that has centralised the role of the teacher is at odds with the interest in this topic and the need for theory-driven guidance suited to teachers. Research is reported that aimed to support the provision of outdoor learning opportunities for children aged between three and eleven at nine settings (pre- or primary schools) in Scotland and England. A set of flexible guidelines that link theory and practice were used by 14 teachers. This article focuses on teachers’ reports of their activity. The incorporation of outdoor learning within conventional teaching and learning approaches is found to be feasible and the implications for practice and future research are explored.     

Why and how mathematicians read proofs: further evidence from a survey study

In a recent paper (Weber & Mejia-Ramos, Educational Studies in Mathematics, 76, 329–344, 2011), we reported findings from two small-scale interview studies on the reasons why and the ways in which mathematicians read proofs. Based on these findings, we designed an Internet-based survey that we distributed to practicing mathematicians working in top mathematics departments in the USA. Surveyed mathematicians (N?=?118) agreed to a great extent with the interviewed mathematicians in the exploratory studies. First, the mathematicians reported that they commonly read published proofs to gain different types of insight, not to check the correctness of the proofs. Second, they stated that when reading these proofs, they commonly: (a) appeal to the reputation of the author and the journal, (b) study how certain steps in the proof apply to specific examples, and (c) focus on the overarching ideas and methods in the proofs. In this paper, we also report findings from another section of the survey that focused on how participants reviewed proofs submitted for publication. The comparison of participant responses to questions in these two sections of the survey suggests that reading a published proof of a colleague and refereeing a proof for publication are substantially different activities for mathematicians.     

Formative assessment in an online learning environment to support flexible on-the-job learning in complex professional domains

This article describes a blueprint for an online learning environment that is based on prominent instructional design and assessment theories for supporting learning in complex domains. The core of this environment consists of formative assessment tasks (i.e., assessment for learning) that center on professional situations. For each professional situation, three levels of situational complexity are defined, and within each of these three levels, tasks are offered that differ in the degree of support offered to the learner. This environment can support (beginning) professionals in complex domains in gaining insight into the available repertoire of behavior in professional situations, as well as into the quality and effectiveness of that behavior (assessment criteria), while simultaneously helping them to develop insight into the standards that their own behavior should (eventually) match.     

Qualitative differences in the exploration of upright and upside-down faces in four-month-old infants: an eye-movement study

Four-month-old infants were habituated with an upright or an upside-down face. Eye-movement recordings showed that the upright and upside-down faces were not explored the same way. Infants spent more time exploring internal features, mainly in the region of the nose and mouth, when the face was upright. They also alternated as frequently between the face's internal features (eyes vs. nose/mouth) as between external and internal features. When the face was upside down, the infants spent half of their time exploring external features, and preferentially alternated between external features and internal features. The main effect of inversion was a decrease of the looking time to the nose/mouth region and of the number of shifts between the eye region and the nose/mouth region.     

Collaborative problem-solving with a computer: How can an interactive learning environment be designed?

This study was designed to evaluate the effects of a «familiar» learning situation on the strategies used by students in a computer mediated interactive environment. We examine the strategies used by 14–15 year-old students in solving physics problems according to whether the technical situation was presented using a canonical version of an electric circuit (rules currently used in class) or a non-canonical version. Students worked either in pairs or individually with a computer. The results show that the dyads only performed better than individuals in the non-canonical version to the problem. The results are discussed in a perspective which offers a renewed approach of student-computer interaction.     

A study of handwriting in written stories of normal and learning disabled children

Written stories of normally achieving and learning disabled children in grades one through three were compared, using a Handwriting Evaluation Scale designed for this study. The subjects also were given tests for receptive language, figure copying and spelling. The Non-LD and LD groups differed on figure copying, spelling and written productivity, but not receptive language. The Non-LD grade level groups differed significantly on two components of the handwriting scale (Letter Size and Control), while the LD grade level groups differed on three components (Letter Formation, Alignment and Spacing, and Letter Size). The most pronounced differences between the LD and normally achieving children were on Formation and Size. A separate analysis of the third grade stories revealed that handwriting was less related to productivity than spelling and visual-motor skills. Nevertheless, the results indicated that many LD students have weak visual-spatial-motor skills. Implications for intervention are discussed.