Provoking contingent moments: Knowledge for ‘powerful teaching’ at the horizon

Background: Teacher knowledge continues to be a topic of debate in Australasia and in other parts of the world. There have been many attempts by mathematics educators and researchers to define the knowledge needed by teachers to teach mathematics effectively. A plethora of terms, such as mathematical content knowledge, pedagogical content knowledge, horizon content knowledge and specialised content knowledge, have been used to describe aspects of such knowledge.

Purpose: This paper proposes a model for teacher knowledge in mathematics that embraces and develops aspects of earlier models. It focuses on the notions of contingent knowledge and the connectedness of ‘big ideas’ of mathematics to enact what is described as ‘powerful teaching’. It involves the teacher’s ability to set up and provoke contingent moments to extend children’s mathematical horizons. The model proposed here considers the various cognitive and affective components and domains that teachers may require to enact ‘powerful teaching’. The intention is to validate the proposed model empirically during a future stage of research.

Sources of evidence: Contingency is described in Rowland’s Knowledge Quartet as the ability to respond to children’s questions, misconceptions and actions and to be able to deviate from a teaching plan as needed. The notion of ‘horizon content knowledge’ (Ball et al.) is a key aspect of the proposed model and has provoked a discussion in this article about students’ mathematical horizons and what these might comprise. Together with a deep mathematical content knowledge and a sensibility for students and their mathematical horizons, these ideas form the foundations of the proposed model.

Main argument: It follows that a deeper level of knowledge might enable a teacher to respond better and to plan and anticipate contingent moments. By taking this further and considering teacher knowledge as ‘dynamic’, this paper suggests that instead of responding to contingent events, ‘powerful teaching’ is about provoking contingent events. This necessarily requires a broad, connected content knowledge based on ‘big mathematical ideas’, a sound knowledge of pedagogies and an understanding of common misconceptions in order to be able to engineer contingent moments.

Conclusions: In order to place genuine problem-solving at the heart of learning, this paper argues for the idea of planning for contingent events, provoking them and ‘setting them up’. The proposed model attempts to represent that process. It is anticipated that the new model will become the framework for an empirical research project, as it undergoes a validation process involving a sample of primary teachers.     

中国耕地警戒值的测算与讨论

耕地安全是食物安全的基础，我国未来的耕地压力大，耕地保护是关键。耕地警戒值是影响政府土地管理部门加强耕地保护执法力度和老百姓建立耕地安全意识、参与性保护耕地的重要指标。近年来，人均耕地警戒值0.053hm²的引用对耕地安全意识的建立产生了很大影响，科学测算我国耕地警戒值具有紧迫性和现实意义。构建最小人均耕地面积的扩展模型S_min=MB∑ni=1βi·GiPi，建立饲料粮食消费量与肉蛋奶消费量、食品作物单产量与单产影响因素之间关系的5个线性回归方程，以及单产影响因素与时间关系等6个曲线估计方程，测量粮食、植物油、糖和蔬菜等食品作物单产量，计算2006年～2015年最小人均耕地面积并与相应年份实际人均耕地面积对比，确定2006年的最小人均耕地面积0.09035hm²和最小人均耕地面积总量1.18×10⁸hm²（17.7亿亩）分别为人均耕地警戒值和耕地总量警戒线。     

Transnational comparisons of teacher expectation of mathematical functional ability in early years and Key Stage 1 pupils – a study undertaken in Hong Kong and England

The dominance of Asian communities in international league tables such as those produced by The Program for International Student Assessment, which monitors competence in literacy and numeracy, has resulted in a policy discourse from the English government of failing schools, deficit curricular and ineffective teachers. This in turn has led to a re-defining of the National Curriculum framework for pupils educated in England with greater focus given in the Early Years curriculum to Mathematical functions particularly those associated with skills in counting; understanding and using number; calculating simple addition and subtraction problems and the understanding of shape, space and measure. Drawing on research carried out in Hong Kong and England this article offers a cross-cultural comparison of the expectations of Early Years Teachers in terms of the capabilities of early learners in achieving elements of number competency. The study of teachers’ expectation is a neglected field in research when applied to compare intercultural achievement. In identifying divergent national expectations, the current panic over what constitutes an appropriate maths curriculum is critiqued and recommendations made for further exploration into teachers’ expectations as to the age and stage at which elements of the maths curriculum are introduced in English schools.     

依托数学建模竞赛,促进数学教学改革

通过开展大学生数学建模竞赛活动,促进了我校高等数学教学改革的不断深入,加强了数学教师与专业教师的沟通与交流,丰富了教学形式和教学内容,激发了同学们的学习兴趣,培养了学生创新意识和竞争能力,提高了我校高等数学教学质量。     

安全网络数学计算协议

讨论了网络数学计算框架 IAMC的安全性问题,给出了一个用安全协议 SSL/TIS提高数学计算协议MCP安全性的实现方案.改进后的网络数学计算框架可有效地提供计算数据的机密性、完整性和用户认证等安全功能.     

确定技术优先级的数学模型

国家就哪些技术应重点发展,哪些技术应限制发展作出正确决策,对于合理利用有限资源是至关重要的.本文基于对影响战略层次上的技术优先级的4个要素的分析,提出一种数学方法,可以有效地解决技术优先级的确定问题.本文采取的4个要素的度量方法,实质上是从定性思维向定量评估的合理转化.本方法既实用,又便于操作.     

全程跟踪卫星或飞船的测控站数量模型

为了确定全程跟踪卫星或飞船的测控站数量,在不考虑地球自转影响和考虑地球自转影响的情况下,使用几何方法分别建立了对卫星或飞船进行全程测控时所需要的最少测控站数量模型,并针对＂神舟七号＂飞船给出了所需最少测控站数量,估算出＂神舟七号＂飞船所能测控的有效范围。     

数学分析教材研究(二)

本文阐述了数学分析课程,数学分析课程教学内容,数学分析课程教学目标,数学分析课程教学方手段,数学分析课程教学理念,数学分析教与学的基本关系;研究了两套历史悠久的数学分析沃土教材的独特风采、独异风格。     

数学分析课程的教学改革与实践

建设一批精品课程是深化教学改革、提高教学质量的一项重要举措。为提高数学分析精品课程教学质量,应该加强师资队伍建设,重视教学内容更新与改革,努力培养学生独立思考问题和解决问题的能力,同时在教学方法上要采用现代教学手段,充分利用网络和多媒体教学。     

信息与计算科学专业《数学分析》课程实践教学探索

文章通过目前《数学分析》课程的教学现状,针对信息与计算科学专业《数学分析》课程进行实践教学探索。在《数学分析》课程中引入数学软件教学,同时通过对教学效果进行研究分析,提出加强数学软件实践、改革教学方式、改进教学手段的必要性,并对实践教学改革提出可行性建议。