In Pursuit of a Connected Way of Knowing: The Case of One Mathematics Teacher

In this paper, we offer illustrations of a mathematics teacher’s difficulties with content knowledge when trying to find connections between school mathematics and science; we do so by describing the development of this teacher’s thinking and learning in her pursuit of connections between the concepts of slope of a line and density of matter. The paper is based on a sub-study that is part of a larger Colombian project, PROMESA (Creating Science and Mathematics Connected Learning Experiences that Open Opportunities for the Promotion of Algebraic Reasoning), which incorporated a Professional Learning Programme (PLP) seeking to integrate school science and mathematics teachers into working teams, in order to create science and mathematics connected learning experiences that considered the promotion of algebraic reasoning. The ‘challenging questions’ that emerged for this teacher, during the workshops of the induction stage of the PLP, became the driving force for her continued engagement in learning mathematics content in a connected way, as opposed to the compartmentalised content-item thinking that she had experienced as a school student. We provide illustrations of first steps in the development of a teacher’s mathematical understanding, which can support growth of mathematical knowledge for teaching.     

数学理解的三种基本形态及教学启示

数学理解包括三种基本形态,即:记忆性理解、解释性理解和探究性理解,这三种数学理解分别对应着“记得、晓得和明得”三种不同的状态。三种数学理解对数学学习都是有价值的,但仅有记忆性和解释性理解是不够的,探究性理解才是数学教学的最终目标。实践中,不少水平不高的教师常常只能让学生达到记忆性理解,有一定水平的教师能让学生达到解释性理解,真正让学生达到探究性理解的教师并不是很多。教师要不失时机地促进学生数学理解层级的迭代升级,促使学生最终达到探究性理解,吴文俊院士数学学习的经验对把握数学理解的三种基本形态有借鉴和启迪意义。在课堂教学中引导学生从事生动活泼的数学探索性活动常常是一个相当艰难的过程,对教师的数学探究素质提出了较高的要求,教师应努力引导学生去探求数学知识的意义和发现的过程,促使学生数学探究性理解方式的养成。     

Using digital technologies in mathematics teaching: developing an understanding of the landscape using three “grand challenge” themes

This paper develops an understanding of the issues, interests and concerns within the mathematics education community related to the use of computers and other digital technologies in the teaching and learning of mathematics. It begins by arguing for the importance of understanding this landscape of interests and concerns, and then turns to the theoretical and methodological choices made in this study, explaining how it has drawn on the approach developed by the STELLAR European Network of Excellence. By analysing the titles and abstracts of a conference chosen to represent the mathematics education community, it maps out the landscape framed by three “Grand Challenges”, finding that an understanding of orchestrating learning is at the heart of the interests of the community, and that the community is interested in exploring new and different contexts for the teaching and learning of mathematics. However, there is currently less interest in investigating and exploiting the increasing connectedness of learners within this community. Further, while the “Grand Challenges” framing is useful in mapping the landscape, it fails to take into account both the personal concerns of teachers and students, such as attitude and confidence, and issues related to doing research and understanding research concerns.     

Conceptual change in mathematics: From rational to (un)real numbers

From an educational point of view, mathematics is supposed to have a completely hierarchical structure in which all new concepts logically follow from prior ones. In this article we try to show that there are also concepts in mathematics which are difficult to learn because of problematic continuity from prior knowledge to new concepts. We focus on the problems of conceptual change connected with the learning of calculus and the shift from rational to real numbers. We demonstrate the difficulty of this conceptual change with the help of historical and psychological evidence. In the empirical study 65 students of higher secondary school were tested after a 40 hour calculus course. In addition, 11 students participated in individual interview. According to the results the conceptual change from a discrete to a continuous idea of numbers seems to be difficult for students. None of the subjects had developed an adequate understanding of real numbers although they had learned to carry out algorithmic procedures belonging to calculus. We discuss how appropriate recent theoretical ideas on conceptual change are for explaining learning problems in this domain. Also some educational implications are presented.     

Knowing more about things you care less about: Cross-sectional analysis of the opposing trend and interplay between conceptual understanding and interest in secondary school chemistry

The development of students' interest in school science activities, their understanding of central chemical concepts, and the interplay between both constructs across Grades 5–11 were analyzed in a cross-sectional paper-and-pencil study (N = 2,510, mean age 11–17 years). Previous empirical findings indicate that students' knowledge increases over the time of secondary school while students' interest, especially in natural science subjects, tends to decrease. Concomitantly, there is evidence for an increase in the positive coupling between interest and knowledge across time. However, previous studies mainly rely on rather global measures, for example, school grades or general subject-related interest, and focus on science as an integrated subject instead of specific disciplines, for example, chemistry. For this article, more proximal and differentiated measures for students' understanding of three chemical concepts (Chemical Reaction, Energy, Matter) and interest in seven dimensions of school science activities according to the RIASEC + N model (Realistic, Investigative, Artistic, Social, Enterprising, Conventional, and Networking; cf. Dierks, Höffler, & Parchmann, 2014) were applied. The results are in line with previous research indicating a general increase in conceptual understanding and a decline in students' interest for all school science activities. However, the interplay between conceptual understanding and interest differs across the seven dimensions. Interest in activities which are likely to promote cognitive activation (investigative, networking) or involving the communication of knowledge (social, enterprising, and networking) are increasingly connected to conceptual understanding, especially in upper secondary grades. Interest in guided hands-on activities (realistic) which are typical in secondary science teaching, however, shows only small positive correlations to students' conceptual understanding across all grades. Hence, in upper-secondary school, investigative, social, enterprising, and networking activities seem to provide opportunities to benefit most from the interrelation between students' interests and their understanding.     

论中小学生的数学观

中小学生的数学观包括数学知识观、数学学习观和数学自我概念。它是通过学生自身数学实践活动经验、教师的教学目标和过程以及社会文化与学校文化传统三方面交互作用的过程形成的。它对学生的数学学习行为、学习策略、动机与情感都会产生重要影响 ,从而对良好数学学习成绩的获得有重要作用     

Developing a deeper understanding of <Emphasis Type="Italic">mathematics teaching expertise</Emphasis>: an examination of three Chinese mathematics teachers’ resource systems as windows into their work and expertise

In order to develop a deeper understanding of mathematics teaching expertise, in this study we use the Documentational Approach to Didactics to explore the resource systems of three Chinese mathematics “expert” teachers. Exploiting the Western and Eastern literature we examine the notion of “mathematics teaching expertise”, as it is perceived in the East and the West. The data consist of two rounds of in-depth interviews, observations and teachers’ representations of their resource systems, where teachers describe their resources connected to their practice, their perceptions of mathematics teaching expertise, and how to develop it. Subsequently, the data are analyzed with respect to the different facets of the notion of teaching expertise and related to the teachers’ views and practices, in order to deepen our understandings of what proficiency in mathematics teaching might mean and how to develop it, seen through the lens of ‘resources’. The significance of the study relates to the enhancement of mathematics teachers’ expertise and capacity building when working in collectives (e.g., in teacher professional development), in order to develop a strong workforce for supporting and helping to improve pupil learning.     

事实性知识、概括性知识与“大概念”--以语文学科为背景

林恩·埃里克森建立的“知识的结构”模型,很好地解释了以“大概念”组织单元的原理和机制。该模型分两个层面:第一层面是事实性知识,有“事实”和“主题”两个层级;第二层面是概括性知识,主要有“概念”和“概括性理解”两个层级。在教学中需依赖具体的事实性知识去发现或获得某一概念,经由理解某一概念构成一种“概念性视角”,凭借“概念性视角”去处理相应主题的具体事实。在两个层面相互作用的认知探究过程中,建立某一概念与其他概念的联系。“概念性理解”就是“由事实性实例支撑的真理”,可称为“概括性知识”。从学习内容的角度,“大概念”实际上是跨学科或学科“核心的概括性知识”。     

我国中小学数学课程中“问题提出”的演变——基于课程标准(教学大纲)的分析

中小学数学课程标准(教学大纲)中对“问题提出”的相关要求影响教科书的编写、教师的教和学生的学。采用内容分析法,对10份涉及“问题提出”的中小学数学课程标准(教学大纲)进行编码,通过定量、定性统计分析发现,中小学数学课程文件中对“问题提出”重视程度逐渐提高,建构起了从宏观到微观的“问题提出”体系;内容表述由单一逐步转向综合;目标要求由模糊笼统趋向于明确具体。未来还需要提高数学课程标准中“问题提出”的实践性;增强课程标准和教科书中“问题提出”的一致性;促进教师成为好的“问题提出”者。     

Learning to lose in the mathematics classroom: a critique of traditional schooling practices in the UK

“Back‐to‐basics” policies fit easily within many mathematics classrooms. Features such as order, control, rule following, uniformity, and conformity are relatively commonplace and are now more generally in demand with the current political backlash against progressive forms of schooling. This paper examines some of the influences of these traditional features of classrooms through a case study of one mathematics department in the United Kingdom. Interviews with students, lesson observations, results of questionnaires, and other case study data are analysed to show the ways in which traditional forms of learning can inhibit understanding, reify the divide between school and the “real world,” and suppress the transfer of knowledge. The findings suggest that two key elements of British Conservative education policy the commitment to traditional forms of schooling and the drive to produce school leavers capable of adapting to the demands of the 21st century — are fundamentally in conflict.     

Students’ and teachers’ perceptions of classroom learning environment in Bhutanese eighth-grade mathematics classes

This paper reports a study of students’ and teachers’ perceptions of their classroom learning environment in Bhutanese eighth-grade mathematics classes. Research suggests that positive perceptions of the learning environment can have a positive influence on students’ learning outcomes, interest and engagement in classroom activities. The study was conducted in 2013, using the survey samples of 608 students and 98 teachers from 22 lower- and middle-secondary schools in western Bhutan. Students’ and teachers’ perceptions of the classroom environment were measured using the Mathematics Classroom Learning Environment Survey (MCLES). Students and teachers mostly perceived their classroom environments favourably on the MCLES scales irrespective of gender, school level and school location. The study is significant for understanding and evaluating the implementation of new mathematics curriculum in Bhutanese schools because it could guide the development of strategies for more-productive mathematics classroom learning. It is also significant from the perspective of Bhutan’s national goal of Gross National Happiness because perceptions and happiness always go hand-in-hand.     

The Nature of Social Practice Among School Professionals: Consequences of the Academic Pressure Exerted by Teachers in Their Teaching

The purpose of this study was to measure teachers' views about trust between teachers, trust between the principal and teachers, peer collaboration, positive attitudes towards the school and how these antecedents influence the academic pressure teachers put on pupils with respect to learning and learning intensity and performance. The methodology involved was a cross-sectional survey of 234 teachers from 11 Norwegian schools. The structural equation analysis indicated that principal-teacher trust has a moderately high impact on such constructs as “teacher-teacher trust” and “academic pressure” and that “teacher-teacher trust” has a moderately high impact on teachers' “peer collaboration”. “Peer collaboration” has a lower impact on “academic pressure”, while the impact of “positive attitudes towards the school” was moderately high. The article concludes with a discussion of the knowledge basis for understanding how social practice among teachers and school leaders in school communities is mobilised for a sustained focus on pupil learning. Implications for practice and directions for future research are discussed.     

新时代教育理念下的课堂教学实践探索——以“概率论与数理统计”课程为例

立德树人是高校教育工作的中心环节。新时代教育理念下,课堂教学要实践"价值塑造+能力培养+知识传授"的教学模式。"概率论与数理统计"课程是工科类院校的一门重要的数学基础课程,知识点丰富,应用性强,育人要素多样化。通过分析概率统计课程特点,从知识传授与价值塑造和知识传授与能力培养两方面阐述了概率统计课程的课堂教学实践,以期为相关课堂教学工作提供借鉴。     

论培养数学学习兴趣促进学生能力发展

小学高年级随着数学知识难度的增加,知识量也逐渐增加,数学知识体系之间形成了很强的关性联,因此学生的数学学习兴趣显得尤为重要。教师在教学中可以融入数学游戏活动,注重生活化数学教学,注重创设良好的学习环境,切实提高学生的学习热情。     

Learning about “Half”: Critical Aspects and Pedagogical Strategies in Designed Preschool Activities

This is an empirical inquiry concerning children’s concept development and early mathematics teaching. The intention is to broaden the understanding of preschool children’s perceptions of the concept “half” (as 1 of 2 equal parts of a whole), in designed mathematics teaching settings. Three teachers working with 4–5-year-old children participate in an in-service program, involving continuous and cooperative reflection and theoretically designed teaching activities. Observations of pedagogical strategies and children’s responses reveal that: children show qualitatively different ways of experiencing the same concept; the ways of experiencing are critical for how the intended learning object is perceived; and the dimensions of the learning object are related to each other, suggesting a hierarchical organization of how to perceive aspects of “half.”     

知识元与问题活性化设计研究

数学教学的目的是把科学形态的数学有效地转化为教育形态的数学知识.引入知识元和问题活性化等新概念,以中学数学为例,对新概念进行解析,对提高教师在知识传授过程中活性化知识的量和质是十分有益的.     

Academic continuity through online collaboration: mathematics teachers support the learning of pupils with chronic illness during school absence

Recent applications of technology to mathematics education have been designed with cognitive and constructivist theoretical perspectives in mind, viewing mathematical learning as the acquisition of knowledge through the construction of meanings and connections between concepts. With the advent of increasingly flexible communication technologies, there is both the need and opportunity to consider how they might be utilised, particularly since emergent socio-cultural theories advocate learning in mathematics as an inherently social activity where understanding is developed and negotiated collaboratively. The need to examine effective technology-facilitated learning arose in the context of a research project, currently underway in a number of secondary schools in the state of Victoria and funded by the Australian Research Council. It is investigating the learning needs of pupils who are absent from school for prolonged or intermittent periods owing to chronic illness yet continue with their school studies. An emerging understanding of the significant difference between computer-mediated contact for mere information exchange and communication for teaching and learning has led to a consideration of socio-cultural perspectives on effective mathematical learning and a focussed investigation of technologies able to facilitate them. Early data have demonstrated the potential of videoconferencing, online whiteboarding and interactive whiteboard application sharing, but which require particular resources, aligned infrastructure and teacher support. This article explores issues surrounding the use of such technologies for collaborative mathematical learning in a context where online interaction is being considered for the learning support of pupils unable to attend school.     

Increasing Content Knowledge and Self‐Efficacy of High School Educators through an Online Course in Food Science

Purdue Univ.'s College of Agriculture developed an Advanced Life Sciences (ALS) program in partnership with several high schools across Indiana. As part of ALS, secondary educators take an introductory food science (FS) course (ALS‐Foods) and teach it at their high school. High school students taking the ALS‐Foods receive dual credit for an introductory course required for all FS majors at Purdue. The goal of this project was to develop an online course to improve content knowledge and self‐efficacy of secondary educators in the field of FS. The course was offered over a 3‐wk period and consisted of 3 learning modules focused on food chemistry, food microbiology, and food processing. Modules included class activities, videos, study questions, and teaching tools. Participants were assessed on content knowledge through written assignments, quizzes, and a final examination. Twenty secondary educators from several states were enrolled. Overall, content knowledge increased significantly (P < 0.05) across all 3 modules after completing the course. Highest scores were in food microbiology/safety (84%), followed by food processing (76%) and food chemistry (70%). A precourse survey indicated that the majority (>80%) of participants felt they had “no‐confidence” to “little‐confidence” in teaching FS concepts related to the 3 modules. Upon completing the course, the confidence level of all participants increased to “some‐confidence” or “complete confidence.” By strengthening the knowledge level of secondary educators, they will be better prepared to teach FS and subsequently, more high school students could be exposed to FS and consider it as a career.     

Exploring mathematics teacher knowledge to explain the gap between theory-based recommendations and school practice in the use of connecting tasks

This study is grounded in the theoretical position that solving problems in different ways creates mathematical connections when learning and teaching mathematics. It acknowledges the central role teachers play in providing students with learning opportunities, and it is based on the empirical finding that mathematics teachers are reluctant to solve problems in different ways in the classroom. In this paper we address the contradiction between theory-based recommendations and school mathematics practice. Based on analysis of individual interviews and two group meetings with 12 Israeli secondary school mathematics teachers, we demonstrate that in the context of multiple-solution connecting tasks this discrepancy is caused by the situated nature of the teachers’ knowledge. We also reveal the complex relationship between different types of teacher knowledge and argue the significance of developing a common language between members of the mathematics education community, including teacher educators and researchers. The names of the teachers have been changed to protect their privacy.