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The articles in this special issue are listed below. Click on an article's title to connect to a video produced by the authors of that article.
Discussions of these articles will be held in virtual events hosted by Pat Thompson and Guershon Harel on June 9 and June 10, 2021:
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Ideas Foundational to Calculus Learning and Their Links to Students’ Difficulties
Abstract: Existing literature reviews of calculus learning made an important contribution to our understanding of the development of mathematics education research in this area, particularly their documentation of how research transitioned from studying students’ misconceptions to investigating students’ understanding and ways of thinking per se. This paper has three main goals relative to this contribution. The first goal is to offer a conceptual analysis of how students’ difficulties surveyed in three major literature review publications originate in the mathematical meanings and ways of thinking students develop in elementary, middle and early high school. The second goal is to highlight a contribution to an important aspect that the articles in this issue make but was overshadowed by other aspects addressed in existing literature reviews: the nature of the mathematics students experience under the name “calculus” in various nations or regions around the world, and the relation of this mathematics to ways ideas foundational to it are developed over the grades. The third goal is to outline research questions drawn from these articles for future research. (Click here for article)
The Strange Role of Calculus in the United States
Keywords: Mathematics Education; Calculus; Advanced Placement Calculus; MAA Committee on the Undergraduate Program in Mathematics; Equity
Abstract: Over the past two decades, the number of U.S. students who study calculus while in high school has more than tripled to around 800,000 each year. It is perceived to be a requirement for admission to any of the top universities, regardless of intended major, with increasing pressure to take it ever earlier in the high school curriculum. This has had profound effects on equity of access to higher education, the nature and pacing of preparation for calculus, the quality of instruction in high school calculus, how universities teach single-variable calculus, and how they deal with those students who need mathematics courses that build on this course. (Click here for article)
School Calculus Curriculum and the Singapore Mathematics Curriculum Framework
Keywords: school calculus; informal calculus; applications of calculus; differentiation and integration
Abstract: School calculus plays at least two important roles in the Singapore mathematics curriculum. Firstly, it plays a supporting role for achieving the general education agenda by meeting the aims of mathematics education. Secondly, it prepares students for undergraduate mathematics education. This paper argues how the dual roles are fulfilled by critically examining the Additional Mathematics syllabus for the upper secondary level (students age 15 to 16/17) and the two mathematics syllabuses (H1 and H2 Mathematics) at the pre-university level (students age 17 to 18/19). This paper also studies the response of a group of pre-university students’ performance in a calculus audit that examines students’ concept images about school calculus. The paper concludes with a discussion of the implications of the students’ responses in the calculus audit. (Click here for article)
School Students’ Preparation for Calculus in the United States
Keywords: Transition to Calculus; Mathematical Meanings for Teaching; International Comparisons; Precalculus Textbook Analysis
Abstract: Researchers have been interested in students’ transition to calculus since the late 1800s. One such line of inquiry highlights students’ understandings of high school mathematics as impeding or supporting their successful transition to university mathematics. This paper addresses an underlying question in this line of inquiry: does school mathematics provide opportunities for students to develop productive meanings for calculus? This article reports on U.S. calculus students’ responses to items that assessed students’ variational reasoning, meanings for average rate of change, and representational use of notation—ideas ostensibly addressed in school mathematics. To make sense of students’ difficulty on these items we sought to understand the opportunities students had to reason with these ideas prior to calculus. We use two data sources to understand the likelihood that students had opportunities to construct propitious meanings for function notation, variation, and average rate of change in their secondary mathematics education: meanings for these ideas supported by precalculus textbooks and meanings secondary teachers demonstrated. Our analysis revealed a disconnect between meanings productive for learning calculus and the meanings conveyed by textbooks and held by U.S. high school teachers. We include a comparison of meanings held by U.S. and Korean teachers to highlight that these meanings are culturally embedded in the U.S. educational system and not epistemologically necessary. (Click here for article)
Transition from High School to University Calculus: A Study of Connection
Keywords: Calculus transition; calculus ideas; limit; connection
Abstract: In Tunisia, calculus is a fundamental component of mathematics curriculum in high school and a major requirement at the advanced level in mathematics bachelor’s degree and engineering. Yet, paving the way for a passage between high school calculus and university calculus remains a challenging issue for both mathematics education researchers and practitioners. In this article, we address this issue by focusing on shared features between high school and university expectations in students’ learning of calculus and the main impediment for high school to meet university requirement. Then, we propose an experimental method to help high school teachers reconsidering their actions according to insufficiencies in students’ preparation for university calculus. The result shows that it is possible to find a linkage between high school and university calculus so as to reduce differences and enhance students’ transition. (Click here for article)
What Meanings Are Assessed in Collegiate Calculus in the United States?
Keywords: Calculus; Assessment; Mathematical reasoning; University level mathematics
Abstract: This article presents the results of our analysis of a sample of 254 Calculus I final exams (collectively containing 4,167 individual items) administered at U.S. colleges and universities. We characterize the specific meanings of foundational concepts the exams assessed, identify features of exam items that assess productive meanings, distinguish categories of items for which students’ responses are not likely to reflect their understanding, and suggest associated modifications to these items that would assess students’ possession of more productive understandings. The article concludes with a discussion of what our findings indicate about the standards and expectations for students’ learning of calculus at institutions of higher education in the United States. We also discuss implications for calculus assessment design and suggest areas for further research. (Click here for article)
Teaching Calculus with Infinitesimals and Differentials
Keywords: Calculus; differentials; infinitesimals; nonstandard analysis; definite integral
Abstract: Several new approaches to calculus in the U.S. have been studied recently that are grounded in infinitesimals or differentials rather than limits. These approaches seek to restore to differential notation the direct referential power it had during the first century after calculus was developed. In these approaches, a differential equation like dy = 2 x·dx is a relationship between increments of x and y , making dy / dx an actual quotient rather than code language for the limit of (1/h)[ f ( x + h ) - f ( x )] as h approaches 0. A definite integral of 2 x dx is a sum of pieces of the form 2 x · dx , not the limit of a sequence of Riemann sums. In this article I motivate and describe some key elements of differentials-based calculus courses, and I summarize research indicating that students in such courses develop robust quantitative meanings for notations in single- and multi-variable calculus. (Click here for article)
The Use of Digital Technology to Estimate a Value of Pi: Teachers’ Solutions on Squaring the Circle in a Graduate Course in Brazil
Keywords: Calculus; History of Mathematics; Humans-with-media; GPIMEM.
Abstract: In this research we investigate how mathematics teachers, as graduate students, estimate the value of π by exploring the problem of squaring the circle using digital technology. Initially, we mention some aspects of teaching and learning of cal- culus in the literature, emphasizing studies that use the notion of humans-with-media to highlight the role of technology in mathematical thinking and knowledge production. Insights on the history of mathematics in calculus are also discussed. We developed a qualitative study based on three different solutions created by groups of teachers using the software GeoGebra and Microsoft Excel. All the teachers’ solutions improved the approximation (8/9)2 ≈ π/4, by determining p and q for (p/q)2 ≈ π/4. The first two solutions with GeoGebra emphasized experimentation and visualization, improving the approximation from one to three decimal places. The third solution with Excel pointed out the elaboration of a formula and improved it up to six decimal places. We emphasize how media shaped the strategies and solutions of the groups. Based on these solutions, we explore an approach for cubing the sphere, discussing approximations for π, highlighting the role of media in enhancing conceptual complexity in the solution of mathematical problems. The nature of the strategies for solving a problem is discussed, especially regarding different ways of thinking-with-technology developed by collectives of teachers-with-media. Although we acknowledge an alternative design for the proposed task, the exploration of problems using aspects of the his- tory of mathematics contributes to the state of the art concerning the studies in calculus developed by the Research Group on Technology, other Media, and Mathematics Education in Brazil. (Click here for article)
The Theory of Calculus for Calculus Teachers
Keywords: Calculus; Conceptual Fluidity; Mathematical Existence; Domain of Abstractions; Formal Theory; Infinity
Abstract: This article aims to narrate a significant part of my experience over several semesters teaching fundamental ideas of the theory of Calculus. I intend to explain how my didactic and mathematical mentality were evolving as I witnessed students' conceptual difficulties as they were confronted with situations that generated tensions between their intuitions and the deductive rigor of the theory of calculus. The evolving atmospheres in the classroom, the efforts to teach and the deep obstacles to learning, indicate the existence of an epistemic fissure as one leaves the dynamic and geometric way of thinking and try to adopt a mode of thinking developed from arithmetic rigor. (Click here for article)
Introduction To Calculus Through An Open-Ended Task In The Context Of Speed: Representations And Actions By Students In Action
Keywords: Learning obstacles; mathematical modeling; differential calculus; potential and actual infinity; socially constructed representation.
Abstract: The first calculus course in the province of Quebec (Canada) is taught in the first year of college before university. Statistics show this course is the most difficult one at the collegial level for students and that it prompts many to drop out of school. In the past, among other problems in the learning of calculus, the following two were considered: one related to the need for students' skills to integrate pre-calculus topics and the other related to mathematical infinity, a concept essential to the concepts of limit, derivative, and integral. New trends in mathematical modeling (e.g. as in the Reform of Calculus in the USA and the STEM project) have introduced a new variable to the topic: the use of differential calculus in real contexts. In this paper, we outline students' learning obstacles that relate to mathematical modeling, to the integration of pre-calculus concepts, and to the concept of mathematical infinity. (Click here for article)
Basic Mental Models of Integrals: Theoretical Conception, Development of a Test Instrument, and First Results
Keywords: Grundvorstellung; integral; empirical evidence; approaches in textbooks
Abstract: Four Grundvorstellungen (basic mental models) can be developed about the concept of a definite integral, some of which are used for specific teaching approaches. This article defines the concept of Grundvorstellungen against the background of personal conceptions and concretizes this concept using the integral as an example. The Grundvorstellungen of area, reconstruction, average, and accumulation are presented. A test instrument for measuring the Grundvorstellungen of students is introduced and analyzed in terms of quality criteria. Results relating to Grundvorstellungen of the integral held by first-year students are presented and discussed. (Click here for article)
Keywords: Calculus curriculum; Curriculum revision; Public discourse; South Korea Stakeholders
A Story of the National Calculus Curriculum: How Culture, Research, and Policy Compete and Compromise in Shaping the Calculus Curriculum in South Korea
Abstract: In this paper we examine changes to the national calculus curriculum of South Korea, where mathematics performance often serves as the mark of academic excellence. We describe relevant cultural traditions of mathematics education in South Korea, including the history of curricular changes in calculus at the secondary level. We then investigate how public discourse concerning a calculus curriculum revision is formed through issue framing, and analyze the public discourse among stakeholders in calculus education in South Korea. In addition, we present an analysis of the revised calculus curriculum at the high school level in relation to the issues raised by stakeholder groups. Further, we reflect on how culture, research, and policy can collectively inform curricular changes and educational policies. Last, we discuss the role of mathematics education research in facilitating the balanced formation of public discourse and in supporting teachers' professional autonomy to provide meaningful calculus education to students. The findings suggest a need for building clear boundaries of roles and responsibilities of stakeholders in developing and revising national curricula, and for using research evidence in curriculum reform. (Click here for article)
Designing a High-School Calculus Curriculum – The Case of Israel
Keywords: Calculus; curriculum; Israel; high school; research influence
Abstract: In response to the call for this special issue, we describe the principles and process of designing a high school mathematics curriculum for the highest of three matriculation bound levels in Israel. While the principles and process were the same for all topics included in the curriculum, we focus on calculus and use examples from calculus for illustration. As part of the context in which the calculus component was developed, we present relevant research which the team used, explicitly or implicitly in the design. We also analyze how this research influenced various aspects from the design of entire units down to the formulation of tasks for the students. (Click here for article)
Multivariable Calculus Results in Different CountriesKeywords: Multivariable calculus; functions of two variables; differential calculus; integral calculus
Abstract: This study presents the result of a survey of research studies on the understanding and teaching of multivariable calculus. The goal of this study is to put the results obtained by the authors in studies that took place in Mexico and Puerto Rico in a wider international context. Studies included a wide geographic spectrum including countries in North and South America, Europe, the Middle- East and Eastern Asia. In spite of the diversity of cultures and diversity in theoretical approaches results obtained are similar and include common difficulties and common suggestions in order to design teaching methodologies that can help students deeply understand two-variables functions and calculus. Most studies point out the need of students’ reflection on three- dimensional space, the importance of the use of graphic and geometrical representations and a thorough work on functions before introducing other calculus topics. Research deals in different ways with the idea of generalization as it considers how students’ knowledge of one-variable calculus provides affordances and constraints to their understanding of multivariable calculus. Some research- based strategies which have been experimentally tested with good results are included. Topics considered include: basic aspects of functions of two variables, differential calculus, integral calculus, and limits and continuity. (Click here for article)
The Learning and Teaching of Multivariable Calculus: A DNR Perspective
Keywords: Multivariable Calculus; DNR-Based Instruction; Inhibitive Shortcut; Catalytic Shortcut
Abstract: The paper presents analyses of multivariable calculus (MVC) learning-teaching phenomena through the lenses of DNR-based instruction, focusing on several foundational calculus concepts, including cross product, linearization, total derivative, Chain Rule, and implicit differentiation. The analyses delineate the nature of current MVC instruction and theorize the potential consequences of the two types of instruction to student learning. One of the crucial parameters differentiating between the two types of instruction is the approach to “shortcuts”: inhibitive shortcuts —those which inhibit knowledge growth—and catalytic shortcuts— those which evolve out of learning trajectories that catalyze knowledge growth.
Among the consequences addressed are: understanding functions in terms of covariational processes; understanding derivative as the best linear approximation; understanding the relation between composition of functions and their Jacobian matrices ; understanding the idea underlying the concept of parametrization ; understanding the rationale underpinning the process of implicit differentiation; thinking in terms of structure, appending external inputs into a coherent mental representation, and making logical inferences. (Click here for article)