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Manuel Santos-Trigo; Matías Camacho-Machín; Fernando Barrera-Mora – ZDM: Mathematics Education, 2024
The aim of this paper is to review recently calculus curriculum reforms and research studies that document what types of understanding students develop in their precalculus courses. We argue that it is important to characterize what difficulties students experience to solve tasks that include the use of foundational calculus concepts and to look…
Descriptors: Mathematics Instruction, Calculus, Barriers, Problem Solving
Oehrtman, Michael; Simmons, Courtney – International Journal of Research in Undergraduate Mathematics Education, 2023
Prior research on students' productive understandings of definite integrals has reasonably focused on students' meanings associated to components and relationships within the standard definition of a limit of Riemann sums. Our analysis was aimed at identifying (i) the broader range of productive quantitative meanings that students invoke and (ii)…
Descriptors: Mathematics Skills, Mathematical Models, Mathematical Concepts, Calculus
Barbara Villatoro – ProQuest LLC, 2023
Authors of calculus texts often include graphs in the text with the intent that the graph depicts relationships described in theorems and formulas. Similarly, graphs are often utilized in classroom lectures and discussions for the same purpose. The author or instructor includes function graphs to represent quantitative relationships and how a pair…
Descriptors: Calculus, Graphs, Concept Formation, Mathematical Concepts
Harel, Guershon – ZDM: Mathematics Education, 2021
The paper presents analyses of multivariable calculus 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…
Descriptors: Calculus, Mathematics Instruction, Teaching Methods, Mathematical Concepts
Reed, Zackery; Tallman, Michael A.; Oehrtman, Michael; Carlson, Marilyn P. – PRIMUS, 2022
We present our analysis of 254 Calculus I final exams from U.S. colleges and universities to identify features of assessment items that necessitate qualitatively distinct ways of understanding and reasoning. We explore salient features of exemplary tasks from our data set to reveal distinctions between exam items made apparent by our analytical…
Descriptors: Calculus, College Mathematics, Mathematical Logic, Mathematics Instruction
Soosloff, Elisa; Huey, Maryann; Alexander, Daniel S. – PRIMUS, 2023
In this reflection of teaching, we describe a series of activities that introduce the Taylor series through dynamic visual representations with explicit connections to students' prior learning. Over the past several decades, educators have noted that curricular materials tend to present the Taylor series in a way that students often interpret as…
Descriptors: Mathematics Instruction, Visual Aids, Prior Learning, Teaching Methods
Bašic, Matija; Milin Šipuš, Željka – International Journal of Research in Undergraduate Mathematics Education, 2022
This study aims to address the teaching of integrals in multivariable calculus concerning the role taken by geometry, specifically, geometrical content dealing with boundaries in integrals that appear as curves and surfaces in R[superscript 2] and R[superscript 3]. Adopting the framework of the Anthropological Theory of the Didactic, we approached…
Descriptors: Mathematics Instruction, Calculus, Geometry, Geometric Concepts
Gabour, Manal – International Journal of Mathematical Education in Science and Technology, 2022
In this article special sequences involving the Butterfly theorem are defined. The Butterfly theorem states that if M is the midpoint of a chord PQ of a circle, then following some definite instructions, it is possible to get two other points X and Y on PQ, such that M is also the midpoint of the segment XY. The convergence investigation of those…
Descriptors: Mathematics Instruction, Computer Software, Secondary School Mathematics, College Mathematics
Reed, Zackery; Tallman, Michael A.; Oehrtman, Michael – PRIMUS, 2023
We offer an analysis of calculus assessment items that highlights ways to evaluate students' application of important meanings and support their engagement in generative ways of reasoning. Our central aim is to identify characteristics of items that require students to apply their understanding of key ideas. We coordinate this analysis of…
Descriptors: Mathematics Instruction, Calculus, Mathematical Concepts, Concept Formation
Boyce, Steven; Grabhorn, Jeffrey A.; Byerley, Cameron – Mathematical Thinking and Learning: An International Journal, 2021
Adolescent and children's concepts of multiplication and fractions have been linked to differences in the number of levels of units they coordinate. In this paper, we discuss relationships between adult students' conceptual structures for coordinating units and their pre-calculus understandings. We conducted interviews and calculus readiness…
Descriptors: Correlation, Calculus, Readiness, Mathematical Logic
García-García, Javier; Dolores-Flores, Crisólogo – International Journal of Mathematical Education in Science and Technology, 2021
Mathematical connections play an important role in achieving mathematical understanding. Therefore, in this article, we report research whose objective was to identify mathematical connections that pre-university students make when they solve problems that involve the derivative and the integral. In this research, we consider a mathematical…
Descriptors: Calculus, Mathematics Instruction, Mathematical Concepts, Concept Formation
Hamdan, May – International Journal of Mathematical Education in Science and Technology, 2019
The literature dealing with student understanding of integration in general and the Fundamental Theorem of Calculus in particular suggests that although students can integrate properly, they understand little about the process that leads to the definite integral. The definite integral is naturally connected to the antiderivative, the area under…
Descriptors: Calculus, Mathematics Instruction, Teaching Methods, Mathematical Logic
Zengin, Yilmaz – Education and Information Technologies, 2022
The study focused on how university students constructed proof of the Fundamental Theorem of Calculus (FTC) starting from their argumentations with dynamic mathematics software in collaborative technology-enhanced learning environment. The participants of the study were 36 university students. The data consisted of participants' written…
Descriptors: Mathematics Instruction, Mathematical Logic, Validity, College Students
Adiredja, Aditya P. – PRIMUS, 2021
The complexity in understanding the [epsilon-delta] definition has motivated research into the teaching and learning of the topic. In this paper I share my design of an instructional analogy called the Pancake Story and four different questions to explore the logical relationship between [epsilon] and [delta] that structures the definition. I…
Descriptors: Mathematics Instruction, College Mathematics, Teaching Methods, Calculus
Cunningham, Daniel W. – International Journal of Mathematical Education in Science and Technology, 2018
Modern calculus textbooks carefully illustrate how to perform integration by trigonometric substitution. Unfortunately, most of these books do not adequately justify this powerful technique of integration. In this article, we present an accessible proof that establishes the validity of integration by trigonometric substitution. The proof offers…
Descriptors: Mathematics Education, Trigonometry, Calculus, Mathematical Concepts