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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:

- June 9, 1600 UTC (9:00 am USA Pacific Daylight Time)
- June 10, 2300 UTC (4:00 pm USA Pacific Daylight Time)

You may register for one or both virtual events at this website. Take note that **registration closes on June 7, 2021.** We will send Zoom links to registered participants on June 7, 2021.

You may post questions or comments regarding the presentations at
this URL. **The comments page will go inactive on June 7, 2021.**

**Ideas Foundational to Calculus Learning and Their Links to Students’ Difficulties**

**Keywords: **

**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)

**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)

**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)

**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

**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 Countries**

**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)