Calculus Textbooks Research Articles

PENGEMBANGAN BUKU AJAR KALKULUS DERIVATIF UNTUK MENINGKATKAN PEMAHAMAN KONSEP MAHASISWA

This research aims to develop valid and practical textbooks to improve students' understanding of concepts. This type of research is development research, namely the development of textbooks with the Plomp model: (1) initial investigation, (2) design, (3) realization/construction, (4) testing, evaluation and revision, and (5) implementation. The subject of this study is a student of the Mathematics Education Study Program of STKIP Andi Matappa. Research instrument: textbook validation sheet. Results of the study: (1) the validity score of the content and reliability of the textbook is included in the high category; (2) This textbook has practical criteria; (3) There is an increase in students' understanding of the concept with the average student being in the medium category. The derivative calculus textbook has met the criteria of valid, reliable and practical to improve the understanding of the concept of STKIP Andi Matappa students.

Primitives of continuous functions via continuous piecewise linear functions

In classical calculus textbooks, the existence of primitive functions of continuous functions is proved by using Riemann integrals. Recently, Patrik Lundström gave a proof via polynomials, based on the Weierstrass approximation theorem. In this note, it is shown that the proof will be easy by using continuous piecewise linear functions.

How do widely-used calculus textbooks introduce the concepts of definite integrals?

This study explores opportunities to learn definite integrals in three widely-used textbooks in the US, Definitions, worked examples, and exercise problems were coded using research based cognitive resources in definite integrals to examine if widely-used textbooks provide students with opportunities to explore how two quantities are related to show area under a curve or other contexts. The results show that initial introduction, worked examples, and exercise problems all provide limited opportunities for students to explore the multiplicative relationship between two quantities (integrand and differential) and adding small pieces to understand definite integrals. Although students may be able to solve integral problems using anti - derivatives, it is possible that they may not understand the definite integral conceptually (knowing why the definite integral gives the area under a curve) or be able expand their understanding to other contexts. Implications of these results are also discussed.

A detailed look at continuity in Calculus textbooks

A detailed look at continuity in Calculus textbooks

Multivariable calculus textbook representation of non-cartesian coordinates: a misalignment between multivariable calculus textbook content and upper-division physics application

Upper-division undergraduate physics coursework necessitates a firm grasp on and fluid use of mathematical knowledge, including an understanding of non-cartesian (specifically polar, spherical and cylindrical) coordinates and how to use them. A limited body of research into physics students’ thinking about coordinate systems suggests that even for upper-division students, understanding of coordinate system concepts is emergent. To more fully grasp upper-division physics students’ incoming understanding of non-cartesian coordinates, the prevalence of non-cartesian content in seven popular Calculus textbooks was studied. Using content analysis techniques, a coding scheme was developed to gain insight into the presentation of coordinate system content both quantitatively and qualitatively. An initial finding was that non-cartesian basis unit vectors were absent in all but one book. A deeper analysis of three of the calculus textbooks showed that cartesian coordinates comprise an overwhelming proportion of the textbooks’ content and that qualitatively the cartesian coordinate system is presented as the default coordinate system. Quantitative and qualitative results are presented with implications for how these results might impact physics teaching and research at the middle and upper-division.

Structuralism in differential equations

Structuralism in philosophy of mathematics has largely focused on arithmetic, algebra, and basic analysis. Some have doubted whether distinctively structural working methods have any impact in other fields such as differential equations. We show narrowly construed structuralism as offered by Benacerraf has no practical role in differential equations. But Dedekind’s approach to the continuum already did not fit that narrow sense, and little of mathematics today does. We draw on one calculus textbook, one celebrated analysis textbook, and a monograph on the Navier–Stokes equation to show structural methods like Dedekind’s have long been central to differential equations, and have philosophically respectable ontology and epistemology.

Exploring criteria for university mathematics teachers’ selection of calculus textbooks

This study delves into the nuanced process of calculus textbook selection by university mathematics lecturers. Drawing from a conceptual framework that encompasses various selection criteria, including culture-driven, student-driven, resource-driven, teacher-driven, mathematics-driven, and constraints-driven factors, the research examines the predominant criteria guiding lecturers’ choices. Data from participating lecturers reveal a multifaceted decision-making process. “Culture-driven selection” emerges as a significant criterion, indicating the influence of curriculum committees. Additionally, educators prioritize “student-driven selection,” valuing accessibility and contextualization, while also favoring textbooks with abundant resources and problems. The study underscores the complexity of textbook selection, shedding light on how instructors navigate educational, commercial, and student-centered considerations. These findings offer insights beneficial for textbook authors and publishers, useful for the creation of resources aligned with educators’ preferences.

Calculus instructors’ perspectives on effective instructional approaches in the teaching of related rates problems

Much research has reported on students’ difficulties with solving related rates problems in calculus. In an effort to generate a resource that could potentially address some of these difficulties from a teaching standpoint, a questionnaire about effective instructional approaches related to the teaching of related rates problems, among other things, was administered to 14 veteran calculus instructors. Analysis of the responses provided by the instructors revealed that all the instructors considered the use of diagrams to be helpful when solving related rates problems. Furthermore, a majority of these instructors noted that introducing a set of steps (i.e., a guideline), during classroom instruction, that students could follow when solving related rates problems is helpful for students when working with this type of problems. These instructors further identified strengths and weaknesses in the way related rates problems are typically presented in calculus textbooks. Implications for instruction are included.

Validity of Derivative Calculus Textbook to Increase Learning Motivation

Validity of Derivative Calculus Textbook to Increase Learning Motivation

The Story of Definite Integrals: A Calculus Textbook Narrative Analysis

There is a plethora of textbooks to choose from. While most textbooks contain the same content, teachers need to understand the textbook they have chosen to support their students’ learning, as textbooks can significantly impact student learning and teacher instruction. When searching for a way to understand the various textbooks, my research question became, How do calculus textbooks tell the story of definite integrals? My research is in response to the never-ending search for how teachers can select and use their resources. Also, this study contributes to understanding the relationships between stories and knowledge construction using Dietiker’s (2015) narrative framework to analyze the written calculus curriculum. I wrote the specific stories of introducing the definite integral as told by five calculus textbooks. Analyzing how the stories developed disclosed the similarities and differences among the texts in teaching definite integrals. The analysis also revealed a variety of avenues to introduce and teach definite integrals, including a focus on the area under the curve, Riemann sums, or the Fundamental Theorem of Calculus, and some implications to the context of physics related to these avenues. These insights provide teachers with a deeper understanding of textbook lessons and the variances that can potentially alter student learning.

Computing indefinite integrals by difference equations

In teaching mathematics to first-year undergraduates, and thus in the appropriate calculus textbooks, the task of calculating an integral that satisfies a specific first-order or second-order recurrence relation often appears. These relations are obtained mainly by applying the method of integration by parts. Calculating such integrals is usually tedious, especially for an integer n > 2, time-consuming, and presents the possibility of making a large number of errors when computing involves multiple iterative steps. In [1], it is shown that in two cases (Theorems 2.1. and 2.3), the process of calculating integrals satisfying first-order recurrence relations can be performed quickly using easily memorised closed-form formulas for corresponding primitive functions. The question can rightly be asked whether there is a faster way to calculate other integrals of this type. In this paper, our goal is to give an affirmative answer to such a question, though without convering all situations. Since each recurrence relation is equivalent to a difference equation of the same order, the calculation of integrals mentioned above can be reduced to solving the corresponding difference equations. Since every first-order or second-order linear difference equation is solvable, it follows that for every integral which can be reduced to a first- order or second-order recurrence formula, it is possible to find corresponding primitive functions directly. Sometimes such a procedure is much faster than iterative solving of the integral. Closed-form formulas for the integrals discussed in the following sections are not unknown (see [2]). However, here our goal is to present the idea of computing indefinite integrals using difference equations. We will discuss it in more detail in Section 2. In Section 3, we discuss the application of the results obtained to calculate several improper integrals and the application of some of them in different sciences. An exciting example of such an application is the integral , which in the case n = 1 is used in the kinetic theory of gases, particularly in the Maxwell-Boltzmann distribution of gas molecules by energies (see Remark 4). Also, we compare the formulas obtained by the method of difference equations with the formulas obtained using Wolfram Alpha software (see Remark 5).

Opportunity to learn about optimization problems provided by undergraduate calculus textbooks: A case study

This study investigated opportunities to learn about optimization problems provided by undergraduate calculus textbooks. To accomplish this, we analyzed examples and practice problems from two calculus textbooks widely used in the teaching of calculus in the United States. Findings of this study indicate that a majority of the problems in both textbooks lack realistic and essential contexts, have matching or missing information, and require a low cognitive demand to solve. Setting up objective functions is either not necessary or it is straightforward for most problems in the two textbooks. In addition, both textbooks provide ample opportunities to interpret critical numbers and extrema in context. Overall, findings of this study suggest the existence of a relationship between known students’ difficulties with solving optimization problems and the opportunities to learn about optimization problems provided by calculus textbooks. Implications for several stakeholders, including calculus textbook authors and calculus instructors are discussed.

Studi didactic transposition: Eksplorasi knowledge to be taught pada limit fungsi

The purpose of this research is to describe the analysis of knowledge to be taught on limit of function. This research is part of the didactic transposition research on the concept of limit of function which consists of four stages. Qualitative research with descriptive method is used in this study. The target is a differential calculus course taught in the first semester at a university in Aceh. Data collection techniques consist of documentation studies and unstructured interviews. Analysis of calculus text book documents, especially on limit functions, was carried out using a praxeology approach. The results of the analysis show that the description of the differential calculus course contains material on the real number system, functions and graphs, limits of functions, continuity of functions, derivatives of functions, and applications of derivatives. This is different from the presentation of material in RPS which is compiled by a team of lecturers supporting differential calculus courses. For presentation of material in calculus textbooks, the concept of the limit of a function begins with an intuitive or informal introduction to the concept. Several cases are given to deepen the concept of limits through graphs, analytics, and numbers.

The role of textbooks for supporting instruction with inquiry: the case of Active Calculus

In this case study, we investigate how an inquiry-oriented interactive calculus textbook supported one U.S. college instructor’s classroom teaching. We utilized analytical inquiry-oriented frameworks on observation data from a small classroom to capture textbooks’ influence on the instruction conceptualized by the Teacher-Mathematics-Student face of the socio-didactical tetrahedron [Rezat, S., & Sträßer, R. (2012). From the didactical triangle to the Rezat and Sträßer’s tetrahedron model socio-didactical tetrahedron: Artifacts as fundamental constituents of the didactical situation. ZDM - Mathematics Education, 44(5), 641–651. https://doi.org/10.1007/s11858-012-0448-4]. Operationalizing specific inquiry-oriented processes allowed us to examine the depth of the interactions conceptualized by the tetrahedron. The instructor’s reliance on the textbooks, their activities, and their supplemental materials created opportunities for meaningful classroom interactions about calculus, supporting a shift towards inquiry. We conclude by suggesting further research on the role of textbooks in supporting undergraduate mathematics instruction with inquiry and discussing implications for practice.

Correction to: Analyzing the written discourse in calculus textbooks over 42 years: the case of primary objects, concrete discursive objects, and a realization tree of the derivative at a point

Correction to: Analyzing the written discourse in calculus textbooks over 42 years: the case of primary objects, concrete discursive objects, and a realization tree of the derivative at a point

Investigation of Rules for Decomposition of the Polynomial of the Form (N(p))/(D(p)) into Partial Fractions.

An investigation of the decomposition of the polynomial of the form (N(p))/(D(p)) into partial fractions is conducted in this article. Two approaches to the decomposition of the fraction are outlined, to enhance the teaching of the concept to students. Many Calculus textbooks give common rules for the decomposition of (N(p))/(D(p)) into Partial Fractions. The rule that if the denominator D(p)=〖(p〗^2-a^2)(bp^2+cp+d), for bp^2+cp+d an irreducible quadratic expression, and that the degree of the numerator N is less than that of the denominator D, that is, deg⁡(N)

Quantitative and covariational reasoning opportunities provided by calculus textbooks: the case of the derivative

While research on the opportunity to learn about mathematics concepts provided by textbooks at the secondary level is well documented, there is still a paucity of similar research at the undergraduate level. Contributing towards addressing this knowledge gap, the present study examined opportunities to engage in quantitative and covariational reasoning, in the context of the derivative concept, provided by two textbooks commonly used in the teaching of calculus in the United States-a regular calculus textbook and an applied calculus textbook. Analysis of expository sections, examples, and exercises related to the derivative provided in these textbooks revealed three main results. First, the applied calculus textbook provides plenty of opportunities to engage in quantitative reasoning. Similar opportunities are limited in the regular calculus textbook. Second, opportunities to engage in covariational reasoning are not only minimal in both textbooks, but also limited to low levels of covariational reasoning, namely coordination, direction, and quantitative coordination. Third, the applied calculus textbook consistently defines ordinary and partial derivatives as rates of change while the regular calculus textbook consistently defines these concepts as limits of a difference quotient. Findings of this research have implications for several stakeholders, including calculus textbook authors and calculus instructors.

사범대학 해석학 전공 교재에서 나타나는 지수함수 미분법의 정당화 방식에 관한 연구

The purpose of this paper is to explore how the mathematical content knowledge on the derivative of the exponential function is demonstrated in analysis courses, compared to the school mathematics where is introduced without rigorous justification. We analyzed structures of justifications of in analysis textbooks used in department of mathematics education at college of education in South Korea, and compared the structures with learning trajectories in school mathematics. As a result, there were four major types of the justifications and one of them fills the gap in the learning trajectories in the high school calculus textbooks. This result suggests the necessity of understanding mathematics content knowledge for pre-service teacher education in connection with school mathematics.

Disciplinary Literacy, Agency, and Didactical Texts: Findings from a Calculus Textbook Think-Aloud Study

Textbooks have been the subject of research within and across disciplines, but have not yet been widely studied from a disciplinary literacy perspective. Readerly agency is also understudied in disciplinary literacy. The present paper aims to illuminate both of these areas by examining facets of agency that readers demonstrated during think-aloud interviews with a college calculus textbook, thus exploring the disciplinary literacies that are involved in reading this didactical text. Authors have drawn on the recently forwarded theory of “didactical disciplinary literacy,” as well as definitions of agency in both literacy and math fields, to reveal how agency was enacted by readers using the textbook to negotiate unknowns in calculus. Findings show participants exhibiting agency in a range of ways, but not always in direct relationship to themselves as agents within the field of mathematics. This study invites college instructors and secondary teachers to consider ways they might foster didactical disciplinary literacy skills to support student learners in mathematics, and other disciplines.

Analyzing the written discourse in calculus textbooks over 42 years: the case of primary objects, concrete discursive objects, and a realization tree of the derivative at a point

Analyzing the written discourse in calculus textbooks over 42 years: the case of primary objects, concrete discursive objects, and a realization tree of the derivative at a point