Understanding and describing mathematical knowledge for teaching: knowledge about proof for engaging students in the activity of proving

This article is situated in the research domain that investigates what mathematical knowledge is useful for, and usable in, mathematics teaching. Specifically, the article contributes to the issue of understanding and describing what knowledge about proof is likely to be important for teachers to have as they engage students in the activity of proving. We explain that existing research informs the knowledge about the logico-linguistic aspects of proof that teachers might need, and we argue that this knowledge should be complemented by what we call knowledge of situations for proving. This form of knowledge is essential as teachers mobilize proving opportunities for their students in mathematics classrooms. We identify two sub-components of the knowledge of situations for proving: knowledge of different kinds of proving tasks and knowledge of the relationship between proving tasks and proving activity. In order to promote understanding of the former type of knowledge, we develop and illustrate a classification of proving tasks based on two mathematical criteria: (1) the number of cases involved in a task (a single case, multiple but finitely many cases, or infinitely many cases), and (2) the purpose of the task (to verify or to refute statements). In order to promote understanding of the latter type of knowledge, we develop a framework for the relationship between different proving tasks and anticipated proving activity when these tasks are implemented in classrooms, and we exemplify the components of the framework using data from third grade. We also discuss possible directions for future research into teachers’ knowledge about proof.

诸城琴曲《关山月》音乐风格研究

诸城琴派的特点是它的山东民间音乐风格,《关山月》是具代表性的曲目。从考察琴曲的渊源人手,通过对音乐本体、演奏技法的分析,以及琴曲风格形成的原因的探究,可以初步发现对《关山月》的山东音乐风格。     

动态语法教学资源的开发和运用

动态语法教学对提高学生理解语言和运用语言能力具有重要作用.为此,从动态语法教学资源的开发和利用着手,探讨动态语法教学的理论和实践,对走出语法教学的误区,开辟语法教学新路具有重要意义.     

吴汝纶与近代安徽教育

吴汝纶虽长期在外仕官讲学,但却时刻心系家乡教育。他为创办安徽古学书院劝捐,建议家乡改书院设学堂,驳斥省城救生局提拨白鹤峰书院租息收入,反对劣绅霸占白鹤峰书院新涨洲地,亲手创办桐城学堂。     

我国近现代学校体育思想的诠释

本文从我国近现代体育思想发展演变来各种思想间的矛盾和冲突,对近现代体育思想发展和演变加以梳理和整修,从理论层面上对其整合构建的过程进行理性的思考。     

Safe spaces? A social-ecological perspective on student perceptions of safety in the environment of the school canteen

ABSTRACT

Background Research suggests that negative peer interactions that compromise student safety and wellbeing often occur in spaces at school that are not easily visible, not adequately monitored, overcrowded and/or relatively unstructured. In a large online survey conducted in Swedish schools by the anti-bullying organisation, Friends, a small proportion of students indicated that they felt unsafe in the school canteen and responded to a question about why they felt unsafe there. As the canteen is often reported to be a space where negative peer interactions, such as harassment, bullying and other forms of school violence, occur, but little is known about why negative peer interactions occur there, we were particularly interested in exploring why some students perceived their school canteen to be an unsafe space.

Purpose Taking a social-ecological perspective, our aim was to investigate why students reported feeling unsafe in the environment of the school canteen.

Method In order to investigate this question, we analysed 1,547 responses from students in Swedish schools in grades 3–6 (ages 9–12). The responses were from the anti-bullying organisation Friends’ online questionnaires from 2011 to 2016. In the responses, students who indicated that they felt unsafe in the school canteen explained why they felt this way. Data were analysed qualitatively, using a six-step thematic approach.

Findings The analysis identified four key themes: Space constraints, Time restrictions, The risk of social blunders and The negative actions of others. We discuss the findings in terms of the macrosystem, exosystem, mesosystem and microsystem.

Conclusions In matters of student safety and wellbeing, we argue that it is not only important to consider the social context, but also how that context is interconnected with environmental and structural elements.     

The multiple logics of school gardening: a ‘return to nature’ or ‘love of labour’?

ABSTRACT

This article explores the history of school gardens in educational projects linked to four scholars at Teachers College (Bigelow, Dewey, Kilpatrick and Carney) during the early twentieth century. It concludes that gardening activities were designed primarily for urban children who lacked experience in farming. The role of gardening in experimental schools in the North is compared with proposals for rural schools in the Midwest and for segregated schools for Black youth in the South. Various logics are identified by analysing the educational arguments used to promote or dismiss school gardening, and the rationales for educating or retaining a certain type of labour force. A combination of these arguments appears in Dewey’s vindication of the emergency production of foodstuffs during the First World War. The different logics point to the paradox connecting the themes of a ‘return to nature’ and ‘love of labour’ as constants in the school gardening discourse.     

公安派形成时间和原因的考察

公安派形成的时间和原因向无定说。该文认为,公安派的形成时间是袁宏道吏吴和解官期间。形成的原因除了袁宏道个人的性格、胆识和才力等作家主体因素外,重要的还有两点：一是宏道周围的一批志同道合者;二是吴中地区的特殊社会环境与文化传统的影响。     

Attendance manipulation and efficiency in Chile’s school voucher system

To improve the quality of education, one can either directly reward performance or introduce school choice, private provision, and demand subsidies. The Chilean voucher scheme combines both approaches: an attendance-related subsidy favors school choice and creates incentives for schools to promote attendance throughout the year. With imperfect monitoring, however, institutions may respond by manipulating performance indicators. By analyzing audit data, we find evidence that a large fraction of Chilean schools – including public schools – over-report attendance, with a higher prevalence among for-profit and under-achieving institutions. Expenditure data suggest that manipulation among for-profit schools seems to follow rent extraction purposes rather than educational goals.     

A school-based intervention on elementary students’ school engagement

Prior research has reported signs of low engagement in the early stages of schooling. The present study assessed the effectiveness of a school-based intervention that promotes cognitive, emotional, and behavioral engagement in elementary school children through a story tool. The study followed a cluster-randomized design with 259 fourth graders nested in 12 classes; the classes, not the individuals, were randomly assigned to an intervention or control group. Both groups were assessed in four waves in two measures for each engagement dimension. Data were analyzed with a multilevel approach. Findings show that the intervention enhanced students’ cognitive, emotional, and behavioral engagement. Still, there is a delay before the intervention program exhibits a beneficial effect.Moreover, gender discrepancies were found. Before the intervention, girls showed higher cognitive and emotional engagement, but boys exhibited higher emotional engagement after the intervention. In addition, current results indicate that the program benefited the boys more than the girls. Finally, there was no evidence that the engagement outcomes differed depending on the parent’s educational level. Findings provide valuable information for future research and educational practice.