A locus for transnational exchanges: European mathematical journals for students and teachers, 1860s–1914

While there were a few mathematical journals aimed at teachers and students as early as the 1840s, it was only in the late 19th century that they became more numerous in Europe. This article is based on the analysis of a corpus of European mathematical journals published between the 1860s and World War I, selected in the first place because they were aimed at high school teachers and high school or/and first two years university students, which are often referred to as “intermediate journals”. All these journals had focused on the teaching of mathematics and, as such, they were shaped by the educational context of the country in which they were published. However, leafing through theses journals, one is struck by the fact that the mathematics they published was in fact highly commensurable, and can see that they were the locus of transnational exchanges on mathematical knowledge. This article shows that several aspects of “internationalisation” were in fact at stake in mathematical journals for students: making knowledge from elsewhere available and of publicizing to the whole world the mathematics produced in one country; making people from different countries collaborate. Finally, it focuses on the effects of transnational exchanges between journals for teachers and students: what was the mathematical knowledge that was circulated through them, and in what respect was it different from that published in other mathematical journals?     

Locally logical mathematics: An emerging teacher honoring both students and mathematics

This case study explores the mathematics engagement and teaching practice of a beginning secondary school teacher. The focus is on the mathematical opportunities available to her students (the classroom mathematics) and how they relate to the teacher's personal capacity and tendencies for mathematical engagement (her personal mathematics). We use a mathematical process-and-action approach to analyze mathematical engagement and then employ the teaching triad—mathematical challenge, sensitivity to students, and management of learning—to situate mathematical engagement within the larger context of teaching practice. The article develops the construct of locally logical mathematics to underscore the cogency of mathematical engagement in the classroom as part of a coherent mathematical system that is embedded within a teaching practice. Contributions of the study include the process-and-action approach, especially in tandem with the teaching triad, as a tool to understand nuances of mathematical engagement and differences in demand between written and implemented tasks.     

结合专业特色的理工科数学课程教学探讨

本文以湖南师范大学物理学专业数学物理方程的教学实践为基础,指出了目前理工科数学课程教学中存在的主要问题,探讨了高等院校数学教学与专业特色相结合的教学方法,并从教学内容、师资要求、教学方法和考核方式等方面提出某些具体改革措施,在教学实践中取得了初步成效．     

Mathematics teaching practices with technology that support conceptual understanding for Latino/a students

We analyze how three seventh grade mathematics teachers from a majority Latino/a, linguistically diverse region of Texas taught the same lesson on interpreting graphs of motion as part of the Scaling Up SimCalc study (Roschelle et al., 2010). The students of two of the teachers made strong learning gains as measured by a curriculum-aligned assessment, while the students of the third teacher were less successful. To investigate these different outcomes, we compare the teaching practices in each classroom, focusing on the teachers’ use of class time and instructional format, their use of mathematical discourse practices in whole-class discussions, and their responses to student contributions. We show that the more successful teachers allowed time for students to use the curriculum and software and discuss it with peers, that they used formal mathematical discourse along with less formal language, and that they responded to student errors using higher-level moves. We conclude by discussing implications for teachers and mathematics educators, with special attention to issues related to the mathematics education of Latinos/as.     

Integrating algebra and proof in high school mathematics: An exploratory study

Frequently, in the US students’ work with proofs is largely concentrated to the domain of high school geometry, thus providing students with a distorted image of what proof entails, which is at odds with the central role that proof plays in mathematics. Despite the centrality of proof in mathematics, there is a lack of studies addressing how to integrate proof into other mathematical domains. In this paper, we discuss a teaching experiment designed to integrate algebra and proof in the high school curriculum. Algebraic proof was envisioned as the vehicle that would provide high school students the opportunity to learn not only about proof in a context other than geometry, but also about aspects of algebra. Results from the experiment indicate that students meaningfully learned about aspects of both algebra and proof in that they produced algebraic proofs involving multiple variables, based on conjectures they themselves generated.     

Teacher listening: The role of knowledge of content and students

In this research report we consider the kinds of knowledge needed by a mathematician as she implemented an inquiry-oriented abstract algebra curriculum. Specifically, we will explore instances in which the teacher was unable to make sense of students’ mathematical struggles in the moment. After describing each episode we will examine the instructor's efforts to listen to the students and the way that these efforts were supported or constrained by her mathematical knowledge for teaching. In particular, we will argue that in each case the instructor was ultimately constrained by her knowledge of how students were thinking about the mathematics.     

Integration of Technology in the Design of Geometry Tasks with Cabri-Geometry

Beginning with the gap in France between the institutional support for the use of technology in mathematics teaching and its weak integration into teacher practice, this paper claims that integrating technology into teaching is a long process. The aim of the paper is to identify and analyse the steps in this integration using as an example the evolution over time (3 years) in the design of teaching scenarios based on Cabri-géomètre for high school students. The analysis indicates that the role played by the technology moved from being a visual amplifier or provider of data towards being an essential constituent of the meaning of tasks and as a consequence affected the conceptions of the mathematical objects that the students might construct.This revised version was published online in September 2005 with corrections to the Cover Date.     

Improving Algebra Preparation: Implications From Research on Student Misconceptions and Difficulties

Through historical and contemporary research, educators have identified widespread misconceptions and difficulties faced by students in learning algebra. Many of these universal issues stem from content addressed long before students take their first algebra course. Yet elementary and middle school teachers may not understand how the subtleties of the arithmetic content they teach can dramatically, and sometimes negatively, impact their students' ability to transition to algebra. The purpose of this article is to bring awareness of some common algebra misconceptions, and suggestions on how they can be averted, to those who are teaching students the early mathematical concepts they will build upon when learning formal algebra. Published literature discussing misconceptions will be presented for four prerequisite concepts, related to symbolic representation: bracket usage, equality, operational symbols, and letter usage. Each section will conclude with research‐based practical applications and suggestions for preventing such misconceptions. The literature discussed in this article makes a case for elementary and middle school teachers to have a deeper and more flexible understanding of the mathematics they teach, so they can recognize how the structure of algebra can and should be exposed while teaching arithmetic.     

The state-of-art in mathematical creativity

Creativity is a topic which is often neglected within mathematics teaching. Usually teachers think that it is logic that is needed in mathematics in the first place, and that creativity is not important and learning mathematics. On the other hand, if we consider a mathematician who develops new results in mathematics. we cannot overlook his/her use of the creative potential. Thus, the main questions are as follows: What methods could be used to foster mathematical creativity within school situations? What scientific knowledge, i.e. research results, do we have on the meaning of mathematical creativity?     

数学优生与差生的素质差异的比较研究

本文通过对初中学生基础素质指标与学习成绩的多元统计分析,探讨了数学差生与优生的素质差异,从不同群体的基础素质对数学学习成绩的影响,研究培养优生与转化差生的不同策略。 本项研究得到世界银行贷款的资助,获96年山东省普通高等学校优秀教学成果二等奖;98年1月通过了国家教委世行贷款办主持的鉴定。99年3月获“1998年度全国师范院校基础教育改革实验优秀成果”二等奖及世界银行贷款“师范教育发展”改革课题实验研究优秀项目二等奖。     

Mathematics standards and curricula in the Netherlands

This paper addresses the question of what mathematics Dutch students should learn according to the standards as established by the Dutch Ministry of Education. The focus is on primary school and the foundation phase of secondary school. This means that the paper covers the range from kindergarten to grade 8 (4~14 years olds). Apart from giving an overview of the standards, we also discuss the standards' nature and history Furthermore, we look at textbooks and examination programs that in the Netherlands both have a key role in determining the intended mathematics curriculum. In addition to addressing the mathematical content, we also pay attention to the way mathematics is taught. The domain-specific education theory that forms the basis for the Dutch approach to teaching mathematics is called “Realistic Mathematics Education” Achievement scores of Dutch students from national and international tests complete this paper. These scores reveal what the standards bring us in terms of students' mathematical understanding. In addition to informing an international audience about the Dutch standards and curricula, we include some critical reflections on them.     

The discursive footprint of learning across mathematical domains: The case of the tangent line

This paper employs the commognitive frame (Sfard, 2008) to investigate how experiences with tangents across mathematical domains leave their marks on students’ subsequent work with tangents. To this aim, I introduce the notion of the discursive footprint of tangents and its characteristics by reviewing how tangents are used across mathematical domains in school textbooks. Manifestations of this footprint were sought in 182 undergraduate mathematics students’ responses to a questionnaire about tangents by labelling their responses and by identifying patterns in the endorsed narratives. Manifestations include the identification of characteristics of sole (and combination of) discourses (geometry, algebra, calculus, mathematical analysis) in student responses. Five themes emerged from the analysis: apparent replication of word use in different narratives; geometry-local hybrid discourse; endorsement of conflicting narratives; enrichment of familiar narratives with new words; and, mathematical analysis as a subsuming discourse. Finally, I discuss the potency of the discursive footprint in research and teaching.     

A Sino-German semi-virtual seminar in mathematics education

In summer 2006 the University of Education in Weingarten, Germany, and East China Normal University, Shanghai, performed a semi-virtual seminar with mathematics students on “Mathematics and Architecture”. The goal was the joint development of teaching materials for German or Chinese school, based on different buildings such as “Nanpu Bridge”, or the “Eiffel Tower”. The purpose of the seminar was to provide a learning environment for students supported by using information and communication technology (ICT) to understand how the hidden mathematics in buildings should be related to school mathematics; to experience the multicultural potential of the international language “Mathematics”; to develop “media competence” while communicating with others and using technologies in mathematics education; and to recognize the differences in teaching mathematics between the two cultures. In this paper we will present our ideas, experiences and results from the seminar.     

数学方法论在数学教学中的应用

本文介绍了运用数学方法论的观点指导数学教学的一种崭新的数学教育方式,即应用数学的发展规律、数学的思想方法和数学中的发现、发明与创新的观点设计数学教学．     

Developing shift problems to foster geometrical proof and understanding

Meaningful learning of formal mathematics in regular classrooms remains a problem in mathematics education. Research shows that instructional approaches in which students work collaboratively on tasks that are tailored to problem solving and reflection can improve students’ learning in experimental classrooms. However, these sequences involve often carefully constructed reinvention route, which do not fit the needs of teachers and students working from conventional curriculum materials. To help to narrow this gap, we developed an intervention—‘shift problem lessons’. The aim of this article is to discuss the design of shift problems and to analyze learning processes occurring when students are working on the tasks. Specifically, we discuss three paradigmatic episodes based on data from a teaching experiment in geometrical proof. The episodes show that is possible to create a micro-learning ecology where regular students are seriously involved in mathematical discussions, ground their mathematical understanding and strengthen their relational framework.     

Some environmental and attitudinal characteristics as predictors of mathematical creativity

There are many things which can be made more useful and interesting through the application of creativity. Self-concept in mathematics and some school environmental factors such as resource adequacy, teachers’ support to the students, teachers’ classroom control, creative stimulation by the teachers, etc. were selected in the study. The sample of the study comprised 770 seventh grade students. Pearson correlation, multiple correlation, regression equation and multiple discriminant function analyses of variance were used to analyse the data. The result of the study showed that the relationship between mathematical creativity and each attitudinal and environmental characteristic was found to be positive and significant. Index of forecasting efficiency reveals that mathematical creativity may be best predicted by self-concept in mathematics. Environmental factors, resource adequacy and creative stimulation by the teachers’ are found to be the most important factors for predicting mathematical creativity, while social–intellectual involvement among students and educational administration of the schools are to be suppressive factors. The multiple correlation between mathematical creativity and attitudinal and school environmental characteristic suggests that the combined contribution of these variables plays a significant role in the development of mathematical creativity. Mahalanobis analysis indicates that self-concept in mathematics and total school environment were found to be contributing significantly to the development of mathematical creativity.     

贯通中学和大学数学教与学 集成造就学生数学素养和创新实践能力

针对如何破解我国从应试教育向素质教育转变步履维艰的难题,本文详细介绍了如何贯通中学和大学数学教与学,集成培养学生数学素养和创新实践能力的具体做法和成效.     

Teacher's reflections on experimenting with technology-enriched inquiry-based mathematics teaching with a preplanned teaching unit

According to previous studies, inquiry-based mathematics teaching enhances learning. However, teachers need support in implementing this type of teaching. In this study, a high school teacher was given a short preplanned inquiry-based mathematics teaching unit that included activities with GeoGebra. The teacher was interviewed after every lesson to explore her reflections after teaching. I analyzed how the teacher described the differences between her regular teaching style and the teaching unit and the pros and cons of the teaching unit. The teacher reflected on the roles of the teacher and students, depth of students’ knowledge, her stance toward the teaching unit, constraints for using this type of teaching approach, and challenges in guiding the students. The results give insights to what kind of reflections on technology-enriched inquiry-based mathematics teaching it is possible to initiate with short preplanned teaching units.     

Mathematical Readiness of Entering College Freshmen: An Exploration of Perceptions of Mathematics Faculty

The National Council of Teachers of Mathematics has set ambitious goals for the teaching and learning of mathematics that include preparing students for both the workplace and higher education. While this suggests that it is important for students to develop strong mathematical competencies by the end of high school, there is evidence to indicate that overall this is not the case. Both national and international studies corroborate the concern that, on the whole, US 12th grade students do not demonstrate mathematical proficiency, suggesting that students making the transition from high school to college mathematics may not be ready for its rigors. In order to investigate mathematical readiness of entering college students, this study surveyed mathematics faculty. Specifically, faculty members were asked their perceptions of average entering students' readiness related to relevant mathematical skills and concepts, and the importance of the same skills and concepts as foundations for college mathematics. Results demonstrated that the faculty perceived that average freshman students are generally not mathematically prepared; further, the skills and concepts rated as highly important — namely, algebraic skills and reasoning and generalization — were among those rated the lowest in terms of student competencies.