Research on calculus: what do we know and where do we need to go?

In this introductory paper we take partial stock of the current state of field on calculus research, exemplifying both the promise of research advances as well as the limitations. We identify four trends in the calculus research literature, starting with identifying misconceptions to investigations of the processes by which students learn particular concepts, evolving into classroom studies, and, more recently research on teacher knowledge, beliefs, and practices. These trends are related to a model for the cycle of research and development aimed at improving learning and teaching. We then make use of these four trends and the model for the cycle of research and development to highlight the contributions of the papers in this issue. We conclude with some reflections on the gaps in literature and what new areas of calculus research are needed.     

An ecological reading of mathematical language in a Grade 3 classroom: A case of learning and teaching measurement estimation

In our work in teacher education and professional development, we aim to help teachers to learn to participate in, and create, classroom ecologies that support students’ learning. In this article we focus on the challenges of developing a classroom ecology that provides mathematical sustenance for students. We pay particular attention to the ways in which classroom language can impede the development of a classroom ecology—one where all students are heard and where knowing is understood as participatory. We present recommendations for teaching practice drawn from an ecological reading of the classroom discourse during a series of lessons on measurement in a Grade 3 classroom.     

Differentiation of teaching and learning mathematics: an action research study in tertiary education

Diversity and differentiation within our classrooms, at all levels of education, is nowadays a fact. It has been one of the biggest challenges for educators to respond to the needs of all students in such a mixed-ability classroom. Teachers’ inability to deal with students with different levels of readiness in a different way leads to school failure and all the negative outcomes that come with it. Differentiation of teaching and learning helps addressing this problem by respecting the different levels that exist in the classroom, and by responding to the needs of each learner. This article presents an action research study where a team of mathematics instructors and an expert in curriculum development developed and implemented a differentiated instruction learning environment in a first-year engineering calculus class at a university in Cyprus. This study provides evidence that differentiated instruction has a positive effect on student engagement and motivation and improves students’ understanding of difficult calculus concepts.     

高师本科数学分析教学改革的研究与实践

以信息社会为特征的新世纪开始提出扫“数学盲”的历史任务 ,而脱盲的标志则是要懂得微积分基础 [1 ] ,然微分与积分是数学分析的主体的结构 .因此 ,2 1世纪的大学数学教育应该搞好数学分析课程的改革与建设 ,开展教学研究 .本文将给出我们在数学分析教学改革的研究与实践 [1 - 3] 的一系列的做法 ,以便请教于同行     

A discrete approach to the concept of derivative

The concept of derivative is a basic concept of calculus. It is closely related to the concept of function, the idea of rate of change, and the limit concept. In recent decades, teaching the concept of derivative in mathematics classrooms has changed: a quite formal approach—closely linked to the teaching of calculus at university and based on the sequence concept—has been transformed to or substituted by a new one. This means working with rates of change, an intuitive access to the concepts of limit and derivative. It includes working with real functions right from the beginning, a great emphasis on graphs, and the use of digital technologies. The meaning of sequences has decreased to a point where they are sometimes no longer even taught in the calculus course. In recent years this concept has been criticized for not developing adequate perceptions of the basic concepts of calculus and not sufficiently preparing the students for scientific courses at university. In this paper we present an alternative discrete step-by-step approach to the basic concepts of calculus by working with sequences and difference sequences, functions defined on \( \mathbb{Z}\) and discrete domains of \( \mathbb{Q}\) , and by subsequently developing the concept of rate of change in a discrete learning environment. The paper is based on general theoretical considerations and empirical investigations by the author and is meant as a contribution to classroom design-research or “design science” (Wittmann, Educ Stud Math 29(4):355–374, 1995).     

The generation and use of graphical examples in calculus classrooms: The case of the mean value theorem

We analyzed video data of five instructors teaching the Mean Value Theorem (MVT) in a first-semester calculus course as part of a broader project investigating how active learning strategies were being implemented and supported in calculus courses. We sought to identify the ways examples of functions that did or did not satisfy the conclusion of MVT were generated and used in instruction. Using thematic analysis, we identified four themes that serve as characterizations of examples, which then allowed for the analysis of trends and patterns. We propose that attention to the generation and use of examples serves as one lens for considering how students can be engaged in the mathematical activity of the classroom, with implications for learning. This work contributes to an evolving notion of what is entailed in students’ active learning of mathematics and the role of the instructor in facilitating active learning opportunities.     

Special Issue Article: A Look at Technology's Role in Professional Development of Mathematics Teachers at the Middle School Level

In last month's issue of School Science and Mathematics, Glenda Lappan addressed the dilemma of supporting teachers in continuing to grow professionally in four domains: Learning more mathematics content; improving their pedagogical skills; improving assessment skills; and addapting the curriculum to the needs of their students. Though it is clear that this kind of learning is best done in the context of what is going on in the classroom, teachers' schedules and other problems make it difficult to implement a coherent classroom based, professional development program. In this article, two more knowledge domains are added to this mix: the ability to (a) use technology and (b) teach with technology effectively. Since the process of learning and teaching is a dynamic one, the author describes his vision of the classroom as a laboratory where teachers get to practice and improve in these six areas and get feedback from an audience of their peers. Reflections are based on a current project in Paterson, New Jersey where the author helps middle school teachers use computer software to improve their mathematics teaching and learning.     

On flipping the classroom in large first year calculus courses

Over the course of two years, 2012--2014, we have implemented a ‘flipping’ the classroom approach in three of our large enrolment first year calculus courses: differential and integral calculus for scientists and engineers. In this article we describe the details of our particular approach and share with the reader some experiences of both instructors and students.     

Teaching values and valued teaching in the mathematics classroom: toward a research agenda

In this paper, we first discuss the teaching of values by focusing on the kinds of values that have been discussed and studied in the other papers in this special journal issue and elsewhere. Then we raise a number of issues about the product-based values in mathematics education, which we identify as teaching values and which can be realized through classroom instruction. In the second section, we discuss the process-based valued teaching methods used to maximize the realization of the teaching values in the classroom. As valued teaching may be perceived differently by different people, in the discussion we analyze how it is seen from both students?? and teachers?? perspectives. We end this paper by discussing a number of methodological issues in studying teaching values and valued teaching as well as offering suggestions for future research.     

Teaching and learning mathematics in the collective

In this paper we analyse and explore teaching and learning in the context of a high school mathematics classroom that was deliberately structured as highly interactive and inquiry-oriented. We frame our discussion within enactivism—a theory of cognition that has helped us to understand classroom processes, particularly at the level of the group. We attempt to show how this classroom of mathematics learners operated as a collective and focus in particular on the role of the teacher in establishing, sustaining, and becoming part of such a collective. Our analysis reveals teaching practices that value, capitalize upon, and promote group cognition, our discussion positions such work as teaching a way of being with mathematics, and we close by offering implications for teaching, educational policy, and further research.     

基于电子信息领域创新型人才培养的微积分教学改革探索

电子信息类高校是培养该领域创新人才的重要基地,作为核心基础课程之一的微积分在其中发挥着重要作用.在建设创新型国家的时代背景下,从创新思维、创新能力和创新综合素质等方面讨论了微积分教学改革在电子信息领域创新型人才培养中的积极作用,并从三个不同视角分析了微积分教学改革所面临的诸多机遇和挑战,最后结合教学实践经历对开展微积分教学改革的具体措施浅谈一些思考与尝试.     

Learning from similarities and differences: a reflection on the potentials and constraints of cross-national studies in mathematics

Research in mathematics education that crosses national boundaries provides new insights into the development and improvement of the teaching and learning of mathematics. In particular, cross-national comparisons lead researchers to more explicit understanding of their own implicit theories about how teachers teach and how children learn mathematics in their local contexts as well as what is going on in school mathematics in other countries. Further, when researchers from multiple countries and regions study collaboratively aspects of teaching and learning of mathematics, the taken-for-granted familiar practices in the classroom can be questioned. Such cross-national comparisons provide opportunities for researchers and educators to probe typical dichotomies such as “high-performing” versus “low performing”, “teacher-centred versus student-centred”, or even “East versus West”, in searching for similarities and differences in educational policies and practices in different cultural contexts.     

Prototype Proofs in Type Theory

The proofs of universally quantified statements, in mathematics, are given as “schemata” or as “prototypes” which may be applied to each specific instance of the quantified variable. Type Theory allows to turn into a rigorous notion this informal intuition described by many, including Herbrand. In this constructive approach where propositions are types, proofs are viewed as terms of λ‐calculus and act as “proof‐schemata”, as for universally quantified types. We examine here the critical case of Impredicative Type Theory, i. e. Girard's system F, where type‐quantification ranges over all types. Coherence and decidability properties are proved for prototype proofs in this impredicative context.     

Teaching ideas and activities for classroom: integrating technology into the pedagogy of integral calculus and the approximation of definite integrals

The purpose of this article is to offer teaching ideas in the treatment of the definite integral concept and the Riemann sums in a technology-supported environment. Specifically, the article offers teaching ideas and activities for classroom for the numerical methods of approximating a definite integral via left- and right-hand Riemann sums, along with midpoint and trapezoidal Riemann sums. The activities demonstrate innovative and creative ways of integrating technology, in particular, GeoGebra dynamic software, into the pedagogy of college-level integral calculus. It also provides, among other things, interesting and original teaching ideas incorporating technology, and an evaluation of these activities by the students themselves who experienced these activities with the GeoGebra dynamic software.     

Implementation of IBL in Europe from an Australasian perspective

In this paper the situation in Australasia with respect to research, curricula and practice in inquiry-based learning in mathematics education is examined. As an organising frame the work of researchers in the Mathematics Education Research Group of Australasia (MERGA) was examined for points of intersection with several of the papers that have arisen from the international PRIMAS project, one of several projects funded by the European Union which follow the recommendations of the Rocard Report (Rocard et al. 2007). Although MERGA services researchers in countries such as Singapore in addition to Australia and New Zealand, a survey of outputs of members showed that the latter two countries had more of a tradition of inquiry-based learning practices and research in mathematics education than did Asian countries under the MERGA umbrella such as Singapore. For this reason the focus here is on mathematics education in the schools in these two countries and the classroom research in them that extends or complements the research or issues raised in other papers in this issue of ZDM.     

Pedagogical alternatives for triple integrals: moving towards more inclusive and personalized learning

This paper is based on the presumption that teaching multiple ways to solve the same problem has academic and social value. In particular, we argue that such a multifaceted approach to pedagogy moves towards an environment of more inclusive and personalized learning. From a mathematics education perspective, our discussion is framed around pedagogical approaches to triple integrals seen in a standard multivariable calculus curriculum. We present some critical perspectives regarding the dominant and long-standing approach to the teaching of triple integrals currently seen in hegemonic calculus textbooks; and we illustrate the need for more diverse pedagogical methods. Finally, we take a constructive position by introducing a new and alternate pedagogical approach to solve some of the classical problems involving triple integrals from the literature through a simple application of integration by parts. This pedagogical alternative for triple integrals is designed to question the dominant one-size-fits-all approach of rearranging the order of integration and the privileging of graphical methods; and to enable a shift towards a more inclusive, enhanced and personalized learning experience.     

青年教师课堂教学观摩竞赛评委水平分析

评价青年教师课堂教学质量取决于青年教师教学水平及评委评判水平。本文就１９９４年山西医学院青年教师教学观摩参赛选手得分对评委评判水平进行了分析     

On Some Properties of the Quaternionic Functional Calculus

In some recent works we have developed a new functional calculus for bounded and unbounded quaternionic operators acting on a quaternionic Banach space. That functional calculus is based on the theory of slice regular functions and on a Cauchy formula which holds for particular domains where the admissible functions have power series expansions. In this paper, we use a new version of the Cauchy formula with slice regular kernel to extend the validity of the quaternionic functional calculus to functions defined on more general domains. Moreover, we show some of the algebraic properties of the quaternionic functional calculus such as the S-spectral radius theorem and the S-spectral mapping theorem. Our functional calculus is also a natural tool to define the semigroup e ^tA when A is a linear quaternionic operator.      

Orchestrating Productive Mathematical Discussions: Five Practices for Helping Teachers Move Beyond Show and Tell

This article presents a teaching experiment on the relationship between informal out-of-school and formal in-school mathematics, and the ways each can inform the other in the development of abstract mathematical knowledge. This study concerns the understanding of some aspects of the multiplicative structure of decimal numbers. It involved a series of classroom activities in upper elementary school, using suitable cultural artifacts and interactive teaching methods. To create a substantially modified teaching/learning environment, new sociomathematical norms (Yackel &; Cobb, 1996) were also introduced. The focus was on fostering a mindful approach toward realistic mathematical modeling, which is both real-world based and quantitatively constrained sense-making (Reusser &; Stebler, 1997). In addition, procedures not commonly used in ordinary teaching activities, such as estimation and approximation processes, were also introduced.     

Exploring informal mathematical products of low achievers at the secondary school level

This article examines the notion of informal mathematical products, in the specific context of teaching mathematics to low achieving students at the secondary school level. The complex and relative nature of this notion is illustrated and some of its characteristics are suggested. These include the use of ad-hoc strategies, mental calculations, idiosyncratic ideas, everyday rather than mathematical language, non-symbolic explanations, visual justifications and common-sense based reasoning. The main argument raised in the article concerns the challenge of valuing informal mathematical products, created by low achievers, and using them within the mathematics classroom as means for advancing such students. The data draws from several research and design projects conducted in Israel since 1991. Selected examples of students’ products, gathered from low-track mathematics classrooms involved in these projects, are presented and analyzed. The analyses highlight various features of such products, and portray the possible gains of teaching approaches that legitimize, and build onwards from, informal products of low achievers.