Assessing Preservice Teachers' Mathematics Cognitive Failures as Related to Mathematics Anxiety and Performance in Undergraduate Calculus

ERIC Educational Resources Information Center

Awofala, Adeneye O. A.; Odogwu, Helen N.

2017-01-01

The study investigated mathematics cognitive failures as related to mathematics anxiety, gender and performance in calculus among 450 preservice teachers from four public universities in the South West geo-political zone of Nigeria using the quantitative research method within the blueprint of the descriptive survey design. Data collected were…

Mathematics for the New Millennium

ERIC Educational Resources Information Center

Gordon, Sheldon P.

2004-01-01

Courses below calculus need to be refocused to emphasise conceptual understanding and realistic applications via mathematical modelling rather than an overarching focus on developing algebraic skills that may be needed for calculus. Without understanding the concepts, students will not be able to transfer the mathematics to new situations or to…

Students' Exploratory Thinking about a Nonroutine Calculus Task

ERIC Educational Resources Information Center

Nabb, Keith

2013-01-01

In this article on introductory calculus, intriguing questions are generated that can ignite an appreciation for the subject of mathematics. These questions open doors to advanced mathematical thinking and harness many elements of research-oriented mathematics. Such questions also offer greater incentives for students to think and reflect.…

Backpropagation and ordered derivatives in the time scales calculus.

PubMed

Seiffertt, John; Wunsch, Donald C

2010-08-01

Backpropagation is the most widely used neural network learning technique. It is based on the mathematical notion of an ordered derivative. In this paper, we present a formulation of ordered derivatives and the backpropagation training algorithm using the important emerging area of mathematics known as the time scales calculus. This calculus, with its potential for application to a wide variety of inter-disciplinary problems, is becoming a key area of mathematics. It is capable of unifying continuous and discrete analysis within one coherent theoretical framework. Using this calculus, we present here a generalization of backpropagation which is appropriate for cases beyond the specifically continuous or discrete. We develop a new multivariate chain rule of this calculus, define ordered derivatives on time scales, prove a key theorem about them, and derive the backpropagation weight update equations for a feedforward multilayer neural network architecture. By drawing together the time scales calculus and the area of neural network learning, we present the first connection of two major fields of research.

Colloquium: Fractional calculus view of complexity: A tutorial

NASA Astrophysics Data System (ADS)

West, Bruce J.

2014-10-01

The fractional calculus has been part of the mathematics and science literature for 310 years. However, it is only in the past decade or so that it has drawn the attention of mainstream science as a way to describe the dynamics of complex phenomena with long-term memory, spatial heterogeneity, along with nonstationary and nonergodic statistics. The most recent application encompasses complex networks, which require new ways of thinking about the world. Part of the new cognition is provided by the fractional calculus description of temporal and topological complexity. Consequently, this Colloquium is not so much a tutorial on the mathematics of the fractional calculus as it is an exploration of how complex phenomena in the physical, social, and life sciences that have eluded traditional mathematical modeling become less mysterious when certain historical assumptions such as differentiability are discarded and the ordinary calculus is replaced with the fractional calculus. Exemplars considered include the fractional differential equations describing the dynamics of viscoelastic materials, turbulence, foraging, and phase transitions in complex social networks.

Women 1.5 Times More Likely to Leave STEM Pipeline after Calculus Compared to Men: Lack of Mathematical Confidence a Potential Culprit.

PubMed

Ellis, Jessica; Fosdick, Bailey K; Rasmussen, Chris

2016-01-01

The substantial gender gap in the science, technology, engineering, and mathematics (STEM) workforce can be traced back to the underrepresentation of women at various milestones in the career pathway. Calculus is a necessary step in this pathway and has been shown to often dissuade people from pursuing STEM fields. We examine the characteristics of students who begin college interested in STEM and either persist or switch out of the calculus sequence after taking Calculus I, and hence either continue to pursue a STEM major or are dissuaded from STEM disciplines. The data come from a unique, national survey focused on mainstream college calculus. Our analyses show that, while controlling for academic preparedness, career intentions, and instruction, the odds of a woman being dissuaded from continuing in calculus is 1.5 times greater than that for a man. Furthermore, women report they do not understand the course material well enough to continue significantly more often than men. When comparing women and men with above-average mathematical abilities and preparedness, we find women start and end the term with significantly lower mathematical confidence than men. This suggests a lack of mathematical confidence, rather than a lack of mathematically ability, may be responsible for the high departure rate of women. While it would be ideal to increase interest and participation of women in STEM at all stages of their careers, our findings indicate that if women persisted in STEM at the same rate as men starting in Calculus I, the number of women entering the STEM workforce would increase by 75%.

Women 1.5 Times More Likely to Leave STEM Pipeline after Calculus Compared to Men: Lack of Mathematical Confidence a Potential Culprit

PubMed Central

Ellis, Jessica; Fosdick, Bailey K.; Rasmussen, Chris

2016-01-01

The substantial gender gap in the science, technology, engineering, and mathematics (STEM) workforce can be traced back to the underrepresentation of women at various milestones in the career pathway. Calculus is a necessary step in this pathway and has been shown to often dissuade people from pursuing STEM fields. We examine the characteristics of students who begin college interested in STEM and either persist or switch out of the calculus sequence after taking Calculus I, and hence either continue to pursue a STEM major or are dissuaded from STEM disciplines. The data come from a unique, national survey focused on mainstream college calculus. Our analyses show that, while controlling for academic preparedness, career intentions, and instruction, the odds of a woman being dissuaded from continuing in calculus is 1.5 times greater than that for a man. Furthermore, women report they do not understand the course material well enough to continue significantly more often than men. When comparing women and men with above-average mathematical abilities and preparedness, we find women start and end the term with significantly lower mathematical confidence than men. This suggests a lack of mathematical confidence, rather than a lack of mathematically ability, may be responsible for the high departure rate of women. While it would be ideal to increase interest and participation of women in STEM at all stages of their careers, our findings indicate that if women persisted in STEM at the same rate as men starting in Calculus I, the number of women entering the STEM workforce would increase by 75%. PMID:27410262

A Historical Survey of the Contributions of Francois-Joseph Servois to the Development of the Rigorous Calculus

ERIC Educational Resources Information Center

Petrilli, Salvatore John, Jr.

2009-01-01

Historians of mathematics considered the nineteenth century to be the Golden Age of mathematics. During this time period many areas of mathematics, such as algebra and geometry, were being placed on rigorous foundations. Another area of mathematics which experienced fundamental change was analysis. The drive for rigor in calculus began in 1797…

Culture Points: Engaging Students outside the Classroom

ERIC Educational Resources Information Center

Fraboni, Michael; Hartshorn, Kevin

2007-01-01

In the typical first-year mathematics course--whether it be calculus or a general education quantitative proficiency course--we struggle to help students see the relevance of mathematics to their own lives. Particularly in a focused course such as calculus, there is a danger that students see mathematics as an isolated subject, with applications…

Examining the Implementation of an Innovative Mathematics Curriculum

ERIC Educational Resources Information Center

Hansen, Heidi Britte

2010-01-01

Reform in mathematics instruction at the college level has been slow to arrive (Dossey, Halvorson, & McCrone, 2008), and many institutions of higher learning still follow the calculus model, while fewer and fewer students need calculus for their chosen areas of study (Ganter & Barker, 2003). Instead, mathematics that is applicable and transferable…

Using Student-Made Games to Learn Mathematics

ERIC Educational Resources Information Center

Gallegos, Irene; Flores, Alfinio

2010-01-01

First-year university students design and play their own games, including board, computer, and other kinds of games, to learn mathematical concepts and practice procedures for their pre-calculus and calculus courses. (Contains 2 tables and 8 figures.)

The Impact of Taking a College Pre-Calculus Course on Students' College Calculus Performance

ERIC Educational Resources Information Center

Sonnert, Gerhard; Sadler, Philip M.

2014-01-01

Poor performance on placement exams keeps many US students who pursue a STEM (science, technology, engineering, mathematics) career from enrolling directly in college calculus. Instead, they must take a pre-calculus course that aims to better prepare them for later calculus coursework. In the USA, enrollment in pre-calculus courses in two- and…

Bridging the Gulf between Formal Calculus and Physical Reasoning.

ERIC Educational Resources Information Center

Van Der Meer, A.

1980-01-01

Some ways to link calculus instruction with the mathematical models used in physics courses are presented. The activity of modelling is presented as a major tool in synchronizing physics and mathematics instruction in undergraduate engineering programs. (MP)

Intra-Mathematical Connections Made by High School Students in Performing Calculus Tasks

ERIC Educational Resources Information Center

García-García, Javier; Dolores-Flores, Crisólogo

2018-01-01

In this article, we report the results of research that explores the intra-mathematical connections that high school students make when they solve Calculus tasks, in particular those involving the derivative and the integral. We consider mathematical connections as a cognitive process through which a person relates or associates two or more ideas,…

Differential Calculus: Concepts and Notation.

ERIC Educational Resources Information Center

Hobbs, David; Relf, Simon

1997-01-01

Suggests that many students with A-level mathematics, and even with a degree in mathematics or a related subject, do not have an understanding of the basic principles of calculus. Describes the approach used in three textbooks currently in use. Contains 14 references. (Author/ASK)

Intra-mathematical connections made by high school students in performing Calculus tasks

NASA Astrophysics Data System (ADS)

García-García, Javier; Dolores-Flores, Crisólogo

2018-02-01

In this article, we report the results of research that explores the intra-mathematical connections that high school students make when they solve Calculus tasks, in particular those involving the derivative and the integral. We consider mathematical connections as a cognitive process through which a person relates or associates two or more ideas, concepts, definitions, theorems, procedures, representations and meanings among themselves, with other disciplines or with real life. Task-based interviews were used to collect data and thematic analysis was used to analyze them. Through the analysis of the productions of the 25 participants, we identified 223 intra-mathematical connections. The data allowed us to establish a mathematical connections system which contributes to the understanding of higher concepts, in our case, the Fundamental Theorem of Calculus. We found mathematical connections of the types: different representations, procedural, features, reversibility and meaning as a connection.

The Relevance of Mathematics: Leaders and Teachers as Gatekeeper for Queensland Senior Calculus Mathematics

ERIC Educational Resources Information Center

Easey, Michael; Gleeson, Jim

2016-01-01

The aim of the larger study, of which this paper is a part, is to investigate the decline in Year 10 male students' participation in senior calculus mathematics courses at an independent boys' school located in metropolitan Queensland. This paper draws on Sealey and Noyes's (2010) relevance framework to conduct document analysis and interviews…

Discovering the Art of Mathematics: Using String Art to Investigate Calculus

ERIC Educational Resources Information Center

von Renesse, Christine; Ecke, Volker

2016-01-01

One goal of our Discovering the Art of Mathematics project is to empower students in the liberal arts to become confident creators of art and imaginative creators of mathematics. In this paper, we describe our experience with using string art to guide liberal arts students in exploring ideas of calculus. We provide excerpts from our inquiry-based…

The calculus of differences: Effects of a psychosocial, cultural, and pedagogical intervention in an all women's university calculus class

NASA Astrophysics Data System (ADS)

Steele, Diana F.; Levin, Amy K.; Blecksmith, Richard; Shahverdian, Jill

2005-10-01

The purpose of this study was to investigate the ways in which a multi-layered women's calculus course influenced the participants' learning of mathematics. This study, conducted in a state university in the Midwestern region of the United States, revealed not only that women in this particular section of calculus were likely to select careers that involved mathematics, but that the focus on peer support, psychosocial issues such as self-confidence, and pedagogy helped the young women overcome gender barriers, as well as barriers of class, poverty, and race. In this article we provide some of the relevant quantitative statistics and relate the stories of two particular women through excerpts from interviews, student artefacts, and participant observation data. We selected these young women because they faced multiple barriers to success in Calculus I and might not have completed the course or taken additional mathematics courses without the support structures that were fundamental to the course.

A Historical Perspective on Teaching and Learning Calculus

ERIC Educational Resources Information Center

Doorman, Michiel; van Maanen, Jan

2008-01-01

Calculus is one of those topics in mathematics where the algorithmic manipulation of symbols is easier than understanding the underlying concepts. Around 1680 Leibniz invented a symbol system for calculus that codifies and simplifies the essential elements of reasoning. The calculus of Leibniz brings within the reach of an ordinary student…

Flipping Freshman Mathematics

ERIC Educational Resources Information Center

Zack, Laurie; Fuselier, Jenny; Graham-Squire, Adam; Lamb, Ron; O'Hara, Karen

2015-01-01

Our study compared a flipped class with a standard lecture class in four introductory courses: finite mathematics, precalculus, business calculus, and calculus 1. The flipped sections watched video lectures outside of class and spent time in class actively working on problems. The traditional sections had lectures in class and did homework outside…

Pushing the Limit: A Class Project

ERIC Educational Resources Information Center

Odafe, Victor U.

2012-01-01

Instructors are constantly struggling to help students understand mathematical concepts as well as the relevance of mathematics to the real world. In calculus, students possess misconceptions of the limit concept. "Pushing the Limit" refers to a semester-long calculus class project that required students to read about, interview calculus…

College Readiness: The Evaluation of Students Participating in the Historically Black College and University Program in Pre-Calculus and the Calculus Sequence

ERIC Educational Resources Information Center

Hall, Angela Renee

2011-01-01

This investigative research focuses on the level of readiness of Science, Technology, Engineering, and Mathematics (STEM) students entering Historically Black Colleges and Universities (HBCU) in the college Calculus sequence. Calculus is a fundamental course for STEM courses. The level of readiness of the students for Calculus can very well play a…

The History of the Calculus

ERIC Educational Resources Information Center

Harding, Simon; Scott, Paul

2004-01-01

Calculus is a mathematical concept that is fundamental to how we understand the world around us. Whether it is in the world of technology, finance, astronomy, sociology, medicine, calculus in one form or another can be found. This brief article describes the origins of calculus in Greece, further developments by Newton and Leibniz, and the…

Polynomial Calculus: Rethinking the Role of Calculus in High Schools

ERIC Educational Resources Information Center

Grant, Melva R.; Crombie, William; Enderson, Mary; Cobb, Nell

2016-01-01

Access to advanced study in mathematics, in general, and to calculus, in particular, depends in part on the conceptual architecture of these knowledge domains. In this paper, we outline an alternative conceptual architecture for elementary calculus. Our general strategy is to separate basic concepts from the particular advanced techniques used in…

The Negative Sign and Exponential Expressions: Unveiling Students' Persistent Errors and Misconceptions

ERIC Educational Resources Information Center

Cangelosi, Richard; Madrid, Silvia; Cooper, Sandra; Olson, Jo; Hartter, Beverly

2013-01-01

The purpose of this study was to determine whether or not certain errors made when simplifying exponential expressions persist as students progress through their mathematical studies. College students enrolled in college algebra, pre-calculus, and first- and second-semester calculus mathematics courses were asked to simplify exponential…

Reading the World with Calculus

ERIC Educational Resources Information Center

Verzosa, Debbie

2015-01-01

It is now increasingly recognized that mathematics is not a neutral value-free subject. Rather, mathematics can challenge students' taken-for-granted realities and promote action. This article describes two issues, namely deforestation and income inequality. These were specifically chosen because they can be related to a range of calculus concepts…

Mathematics Placement at Cottey College.

ERIC Educational Resources Information Center

Callahan, Susan

In response to the large numbers of students who were failing or dropping out of basic algebra and calculus classes, Cottey College, in Missouri, developed a math placement program in 1982 using Basic Algebra (BA) and Calculus Readiness (CR) tests from the Mathematical Association of America's Placement Testing Program. Cut off scores for the…

Developing calculus textbook model that supported with GeoGebra to enhancing students’ mathematical problem solving and mathematical representation

NASA Astrophysics Data System (ADS)

Dewi, N. R.; Arini, F. Y.

2018-03-01

The main purpose of this research is developing and produces a Calculus textbook model that supported with GeoGebra. This book was designed to enhancing students’ mathematical problem solving and mathematical representation. There were three stages in this research i.e. define, design, and develop. The textbooks consisted of 6 chapters which each chapter contains introduction, core materials and include examples and exercises. The textbook developed phase begins with the early stages of designed the book (draft 1) which then validated by experts. Revision of draft 1 produced draft 2. The data were analyzed with descriptive statistics. The analysis showed that the Calculus textbook model that supported with GeoGebra, valid and fill up the criteria of practicality.

A transformative model for undergraduate quantitative biology education.

PubMed

Usher, David C; Driscoll, Tobin A; Dhurjati, Prasad; Pelesko, John A; Rossi, Louis F; Schleiniger, Gilberto; Pusecker, Kathleen; White, Harold B

2010-01-01

The BIO2010 report recommended that students in the life sciences receive a more rigorous education in mathematics and physical sciences. The University of Delaware approached this problem by (1) developing a bio-calculus section of a standard calculus course, (2) embedding quantitative activities into existing biology courses, and (3) creating a new interdisciplinary major, quantitative biology, designed for students interested in solving complex biological problems using advanced mathematical approaches. To develop the bio-calculus sections, the Department of Mathematical Sciences revised its three-semester calculus sequence to include differential equations in the first semester and, rather than using examples traditionally drawn from application domains that are most relevant to engineers, drew models and examples heavily from the life sciences. The curriculum of the B.S. degree in Quantitative Biology was designed to provide students with a solid foundation in biology, chemistry, and mathematics, with an emphasis on preparation for research careers in life sciences. Students in the program take core courses from biology, chemistry, and physics, though mathematics, as the cornerstone of all quantitative sciences, is given particular prominence. Seminars and a capstone course stress how the interplay of mathematics and biology can be used to explain complex biological systems. To initiate these academic changes required the identification of barriers and the implementation of solutions.

A Transformative Model for Undergraduate Quantitative Biology Education

PubMed Central

Driscoll, Tobin A.; Dhurjati, Prasad; Pelesko, John A.; Rossi, Louis F.; Schleiniger, Gilberto; Pusecker, Kathleen; White, Harold B.

2010-01-01

The BIO2010 report recommended that students in the life sciences receive a more rigorous education in mathematics and physical sciences. The University of Delaware approached this problem by (1) developing a bio-calculus section of a standard calculus course, (2) embedding quantitative activities into existing biology courses, and (3) creating a new interdisciplinary major, quantitative biology, designed for students interested in solving complex biological problems using advanced mathematical approaches. To develop the bio-calculus sections, the Department of Mathematical Sciences revised its three-semester calculus sequence to include differential equations in the first semester and, rather than using examples traditionally drawn from application domains that are most relevant to engineers, drew models and examples heavily from the life sciences. The curriculum of the B.S. degree in Quantitative Biology was designed to provide students with a solid foundation in biology, chemistry, and mathematics, with an emphasis on preparation for research careers in life sciences. Students in the program take core courses from biology, chemistry, and physics, though mathematics, as the cornerstone of all quantitative sciences, is given particular prominence. Seminars and a capstone course stress how the interplay of mathematics and biology can be used to explain complex biological systems. To initiate these academic changes required the identification of barriers and the implementation of solutions. PMID:20810949

Simplicial lattices in classical and quantum gravity: Mathematical structure and application

NASA Astrophysics Data System (ADS)

Lafave, Norman Joseph

1989-03-01

Geometrodynamics can be understood more clearly in the language of geometry than in the language of differential equations. This is the primary motivation for the development of calculational schemes based on Regge Calculus as an alternative to those schemes based on Ricci Calculus. The mathematics of simplicial lattices were developed to the same level of sophistication as the mathematics of pseudo--Riemannian geometry for continuum manifolds. This involves the definition of the simplicial analogues of several concepts from differential topology and differential geometry-the concept of a point, tangent spaces, forms, tensors, parallel transport, covariant derivatives, connections, and curvature. These simplicial analogues are used to define the Einstein tensor and the extrinsic curvature on a simplicial geometry. This mathematical formalism was applied to the solution of several outstanding problems in the development of a Regge Calculus based computational scheme for general geometrodynamic problems. This scheme is based on a 3 + 1 splitting of spacetime within the Regge Calculus prescription known as Null-Strut Calculus (NSC). NSC describes the foliation of spacetime into spacelike hypersurfaces built of tetrahedra. These hypersurfaces are coupled by light rays (null struts) to past and future momentum-like structures, geometrically dual to the tetrahedral lattice of the hypersurface. Avenues of investigation for NSC in quantum gravity are described.

Productive and ineffective efforts: how student effort in high school mathematics relates to college calculus success

NASA Astrophysics Data System (ADS)

Barnett, M. D.; Sonnert, G.; Sadler, P. M.

2014-10-01

Relativizing the popular belief that student effort is the key to success, this article finds that effort in the most advanced mathematics course in US high schools is not consistently associated with college calculus performance. We distinguish two types of student effort: productive and ineffective efforts. Whereas the former carries the commonly expected benefits, the latter is associated with negative consequences. Time spent reading the course text in US high schools was negatively related to college calculus performance. Daily study time, however, was found to be either a productive or an ineffective effort, depending on the level of high school mathematics course and the student's performance in it.

A Transition Course from Advanced Placement to College Calculus

ERIC Educational Resources Information Center

Lucas, Timothy A.; Spivey, Joseph

2011-01-01

In the Spring of 2007, a group of highly motivated mathematics graduate students conducted a review of Duke's Calculus curriculum. They focused on two main problems. The first problem is the result of a very positive trend: a growing number of students are earning AP credit for Calculus I in high school. However, this results in Calculus II…

Improving Calculus II and III through the Redistribution of Topics

ERIC Educational Resources Information Center

George, C. Yousuf; Koetz, Matt; Lewis, Heather A.

2016-01-01

Three years ago our mathematics department rearranged the topics in second and third semester calculus, moving multivariable calculus to the second semester and series to the third semester. This paper describes the new arrangement of topics, and how it could be adapted to calculus curricula at different schools. It also explains the benefits we…

A Comparison of Mathematics Teachers' and Professors' Views on Secondary Preparation for Tertiary Calculus

ERIC Educational Resources Information Center

Wade, Carol; Sonnert, Gerhard; Sadler, Philip M.; Hazari, Zahra; Watson, Charity

2016-01-01

This article compares the views of teachers and professors about the transition from secondary mathematics to tertiary calculus. Quantitative analysis revealed five categories where teachers and professors differed significantly in the relative frequency of addressing them. Using the rite of passage theory, the separation and incorporation phases…

Non-Mathematics Students' Reasoning in Calculus Tasks

ERIC Educational Resources Information Center

Jukic Matic, Ljerka

2015-01-01

This paper investigates the reasoning of first year non-mathematics students in non-routine calculus tasks. The students in this study were accustomed to imitative reasoning from their primary and secondary education. In order to move from imitative reasoning toward more creative reasoning, non-routine tasks were implemented as an explicit part of…

Motivation, Volition and Belief Change Strategies to Improve Mathematics Learning

ERIC Educational Resources Information Center

Kim, C.; Keller, J. M.

2010-01-01

The purpose of this study was to investigate the effects of motivation, volition and belief change strategies, implemented with personal and group email messages, on students' attitudes, study habits and achievement in a calculus course for non-mathematics majors. Eighty four undergraduates enrolled in a calculus course received emails over a…

Insights from the MAA National Study of College Calculus

ERIC Educational Resources Information Center

Bressoud, David

2015-01-01

Over the past five years, the Mathematical Association of America, with support from the National Science Foundation, has explored the teaching of mainstream Calculus 1 at the postsecondary level, where by "mainstream" we mean those courses that can be used as part of the prerequisite stream to more advanced postsecondary mathematics. We…

Enhancing Student Writing and Computer Programming with LATEX and MATLAB in Multivariable Calculus

ERIC Educational Resources Information Center

Sullivan, Eric; Melvin, Timothy

2016-01-01

Written communication and computer programming are foundational components of an undergraduate degree in the mathematical sciences. All lower-division mathematics courses at our institution are paired with computer-based writing, coding, and problem-solving activities. In multivariable calculus we utilize MATLAB and LATEX to have students explore…

Bridging a Cultural Gap

ERIC Educational Resources Information Center

Leviatan, Talma

2008-01-01

There has been a broad wave of change in tertiary calculus courses in the past decade. However, the much-needed change in tertiary pre-calculus programmes--aimed at bridging the gap between high-school mathematics and tertiary mathematics--is happening at a far slower pace. Following a discussion on the nature of the gap and the objectives of a…

Rate of Change: AP Calculus Students' Understandings and Misconceptions after Completing Different Curricular Paths

ERIC Educational Resources Information Center

Teuscher, Dawn; Reys, Robert E.

2012-01-01

This study examined Advanced Placement Calculus students' mathematical understanding of rate of change, after studying four years of college preparatory (integrated or single-subject) mathematics. Students completed the Precalculus Concept Assessment (PCA) and two open-ended tasks with questions about rates of change. After adjusting for prior…

A preliminary study of achievement, attitudes toward success in mathematics, and mathematics anxiety with technology-based instruction in brief calculus.

PubMed

Alkhateeb, Haitham M

2002-02-01

This study was designed to compare achievement, attitudes toward success in mathematics, and mathematics anxiety of college students taught brief calculus using a graphic calculator, with the achievement and attitudes and anxiety of students taught using the computer algebra system Maple, using a technology based text book. 50 men and 50 women, students in three classes at a large public university in the southwestern United States, participated. Students' achievement in brief calculus was measured by performance on a teacher-made achievement test given at the end of the study. Analysis of variance showed no significant difference in achievement between the groups. To measure change in attitudes and anxiety, responses to paper-and-pencil inventories indicated significant differences in favor of students using the computer.

A Guided Tour of Mathematical Methods - 2nd Edition

NASA Astrophysics Data System (ADS)

Snieder, Roel

2004-09-01

Mathematical methods are essential tools for all physical scientists. This second edition provides a comprehensive tour of the mathematical knowledge and techniques that are needed by students in this area. In contrast to more traditional textbooks, all the material is presented in the form of problems. Within these problems the basic mathematical theory and its physical applications are well integrated. The mathematical insights that the student acquires are therefore driven by their physical insight. Topics that are covered include vector calculus, linear algebra, Fourier analysis, scale analysis, complex integration, Green's functions, normal modes, tensor calculus, and perturbation theory. The second edition contains new chapters on dimensional analysis, variational calculus, and the asymptotic evaluation of integrals. This book can be used by undergraduates, and lower-level graduate students in the physical sciences. It can serve as a stand-alone text, or as a source of problems and examples to complement other textbooks. All the material is presented in the form of problems Mathematical insights are gained by getting the reader to develop answers themselves Many applications of the mathematics are given

Mathematics preparation for medical school: do all premedical students need calculus?

PubMed

Nusbaum, Neil J

2006-01-01

The premedical student confronts a disparate set of required and recommended courses from the various medical schools to which the student might apply. Students may feel compelled to take courses such as calculus even though most medical schools do not require it and even though it may not be related to either undergraduate academic plans or the core academic needs of the typical future physician. Basic mathematical knowledge--a knowledge of algebra, statistics, and overall numeracy--are each more important for most future physicians than is the traditional calculus course.

What Does It Mean for a Student to Understand the First-Year Calculus? Perspectives of 24 Experts

ERIC Educational Resources Information Center

Sofronas, Kimberly S.; DeFranco, Thomas C.; Vinsonhaler, Charles; Gorgievski, Nicholas; Schroeder, Larissa; Hamelin, Chris

2011-01-01

This article presents the views of 24 nationally recognized authorities in the field of mathematics, and in particular the calculus, on student understanding of the first-year calculus. A framework emerged that includes four overarching end goals for understanding of the first-year calculus: (a) mastery of the fundamental concepts and-or skills of…

The Case for Biocalculus: Design, Retention, and Student Performance

PubMed Central

Eaton, Carrie Diaz; Highlander, Hannah Callender

2017-01-01

Calculus is one of the primary avenues for initial quantitative training of students in all science, technology, engineering, and mathematics fields, but life science students have been found to underperform in the traditional calculus setting. As a result, and because of perceived lack of its contribution to the understanding of biology, calculus is being actively cut from biology program requirements at many institutions. Here, we present an alternative: a model for learning mathematics that sees the partner disciplines as crucial to student success. We equip faculty with information to engage in dialogue within and between disciplinary departments involved in quantitative education. This includes presenting a process for interdisciplinary development and implementation of biology-oriented Calculus I courses at two institutions with different constituents, goals, and curricular constraints. When life science students enrolled in these redesigned calculus courses are compared with life science students enrolled in traditional calculus courses, students in the redesigned calculus courses learn calculus concepts and skills as well as their traditional course peers; however, the students in the redesigned courses experience more authentic life science applications and are more likely to stay and succeed in the course than their peers who are enrolled in traditional courses. Therefore, these redesigned calculus courses hold promise in helping life science undergraduate students attain Vision and Change recommended competencies. PMID:28450445

The impact of instructor pedagogy on college calculus students' attitude toward mathematics

NASA Astrophysics Data System (ADS)

Sonnert, Gerhard; Sadler, Philip M.; Sadler, Samuel M.; Bressoud, David M.

2015-04-01

College calculus teaches students important mathematical concepts and skills. The course also has a substantial impact on students' attitude toward mathematics, affecting their career aspirations and desires to take more mathematics. This national US study of 3103 students at 123 colleges and universities tracks changes in students' attitudes toward mathematics during a 'mainstream' calculus course while controlling for student backgrounds. The attitude measure combines students' self-ratings of their mathematics confidence, interest in, and enjoyment of mathematics. Three major kinds of instructor pedagogy, identified through the factor analysis of 61 student-reported variables, are investigated for impact on student attitude as follows: (1) instructors who employ generally accepted 'good teaching' practices (e.g. clarity in presentation and answering questions, useful homework, fair exams, help outside of class) are found to have the most positive impact, particularly with students who began with a weaker initial attitude. (2) Use of educational 'technology' (e.g. graphing calculators, for demonstrations, in homework), on average, is found to have no impact on attitudes, except when used by graduate student instructors, which negatively affects students' attitudes towards mathematics. (3) 'Ambitious teaching' (e.g. group work, word problems, 'flipped' reading, student explanations of thinking) has a small negative impact on student attitudes, while being a relatively more constructive influence only on students who already enjoyed a positive attitude toward mathematics and in classrooms with a large number of students. This study provides support for efforts to improve calculus teaching through the training of faculty and graduate students to use traditional 'good teaching' practices through professional development workshops and courses. As currently implemented, technology and ambitious pedagogical practices, while no doubt effective in certain classrooms, do not appear to have a reliable, positive impact on student attitudes toward mathematics.

Teacher Questioning in Undergraduate Mathematics: A Collective Case Study

ERIC Educational Resources Information Center

White, Tracy Foote

2016-01-01

This study examines the mathematical questioning of undergraduate Calculus I instructors for the purpose of detailing the ways in which instructors are using their questions. The emphasis is on verbal questions because of their in-the-moment value and ability to get students engaged in discourse. Calculus I is of particular interest because of its…

Students' Conceptual Understanding of a Function and Its Derivative in an Experimental Calculus Course

ERIC Educational Resources Information Center

Habre, Samer; Abboud, May

2006-01-01

Calculus has been witnessing fundamental changes in its curriculum, with an increased emphasis on visualization. This mode for representing mathematical concepts is gaining more strength due to the advances in computer technology and the development of dynamical mathematical software. This paper focuses on the understanding of the function and its…

Anticipating Mathematics Performance: A Cross-Validation Comparison of AID3 and Regression. AIR 1988 Annual Forum Paper.

ERIC Educational Resources Information Center

Bloom, Allan M.; And Others

In response to the increasing importance of student performance in required classes, research was conducted to compare two prediction procedures, linear modeling using multiple regression and nonlinear modeling using AID3. Performance in the first college math course (College Mathematics, Calculus, or Business Calculus Matrices) was the dependent…

What Do Croatian Pre-Service Teachers Remember from Their Calculus Course?

ERIC Educational Resources Information Center

Jukic, Ljerka; Brückler, Franka Miriam

2014-01-01

This paper reports a study on retention of core concepts in differential and integral calculus by examining the knowledge of two pre-service mathematics students. The study is conducted using a mixed method approach and the obtained data were analyzed using theory of three worlds of mathematics. The results showed that having good understanding of…

Recommendations for the Undergraduate Mathematics Program for Students in the Life Sciences. An Interim Report.

ERIC Educational Resources Information Center

Committee on the Undergraduate Program in Mathematics, Berkeley, CA.

This report considers the mathematics required by life science students (those with majors in agriculture and renewable resources, all branches of biology, and medicine) who have successfully completed the usual pre-calculus courses. A core is proposed, to include one year of calculus, some linear algebra, and some probability and statistics.…

The Development of an Individualized Instructional Program in Beginning College Mathematics Utilizing Computer Based Resource Units. Final Report.

ERIC Educational Resources Information Center

Rockhill, Theron D.

Reported is an attempt to develop and evaluate an individualized instructional program in pre-calculus college mathematics. Four computer based resource units were developed in the areas of set theory, relations and function, algebra, trigonometry, and analytic geometry. Objectives were determined by experienced calculus teachers, and…

Productive and Ineffective Efforts: How Student Effort in High School Mathematics Relates to College Calculus Success

ERIC Educational Resources Information Center

Barnett, M.D.; Sonnert, G.; Sadler, P.M.

2014-01-01

Relativizing the popular belief that student effort is the key to success, this article finds that effort in the most advanced mathematics course in US high schools is not consistently associated with college calculus performance. We distinguish two types of student effort: productive and ineffective efforts. Whereas the former carries the…

Calculus Instructors' and Students' Discourses on the Derivative

ERIC Educational Resources Information Center

Park, Jungeun

2011-01-01

Recently, there has been an increasing interest in collegiate mathematics education, especially teaching and learning calculus (e.g., Oehrtman, Carlson, & Thompson, 2008; Speer, Smith, & Horvath, 2010). Of many calculus concepts, the derivative is known as a difficult concept for students to understand because it involves various concepts…

The impact of taking a college pre-calculus course on students' college calculus performance

NASA Astrophysics Data System (ADS)

Sonnert, Gerhard; Sadler, Philip M.

2014-11-01

Poor performance on placement exams keeps many US students who pursue a STEM (science, technology, engineering, mathematics) career from enrolling directly in college calculus. Instead, they must take a pre-calculus course that aims to better prepare them for later calculus coursework. In the USA, enrollment in pre-calculus courses in two- and four-year colleges continues to grow, and these courses are well-populated with students who already took pre-calculus in high school. We examine student performance in college calculus, using regression discontinuity to estimate the effects of taking college pre-calculus or not, in a national US sample of 5507 students at 132 institutions. We find that students who take college pre-calculus do not earn higher calculus grades.

Differentiated Instruction in a Calculus Curriculum for College Students in Taiwan

ERIC Educational Resources Information Center

Chen, Jing-Hua; Chen, Yi-Chou

2018-01-01

Objectives: To explore differentiated instruction within a calculus curriculum. For college students to learn concentration, motivation and the impact of academic achievement; explore the attitudes and ideas of students on differentiated instruction within a calculus curriculum; build up the diversity of mathematics education within varied…

An Introductory Calculus-Based Mechanics Investigation

ERIC Educational Resources Information Center

Allen, Bradley

2017-01-01

One challenge for the introductory physics teacher is incorporating calculus techniques into the laboratory setting. It can be difficult to strike a balance between presenting an experimental task for which calculus is essential and making the mathematics accessible to learners who may be apprehensive about applying it. One-dimensional kinematics…

Transitioning from Introductory Calculus to Formal Limit Conceptions

ERIC Educational Resources Information Center

Nagle, Courtney

2013-01-01

The limit concept is a fundamental mathematical notion both for its practical applications and its importance as a prerequisite for later calculus topics. Past research suggests that limit conceptualizations promoted in introductory calculus are far removed from the formal epsilon-delta definition of limit. In this article, I provide an overview…

Advanced Placement Mathematics Calculus, Grade 12 Curriculum Guide.

ERIC Educational Resources Information Center

Scharf, John; And Others

This document is a guide to the advanced placement program in calculus for grade 12 in the city schools in Warren, Ohio. The program covers analytic geometry, differential and integral calculus of algebraic functions, elementary transcendental functions, and applications of differentiation and integration. The philosophy and aims of the program…

To Math or Not to Math: The Algebra-Calculus Pipeline and Postsecondary Mathematics Remediation

ERIC Educational Resources Information Center

Showalter, Daniel A.

2017-01-01

This article reports on a study designed to estimate the effect of high school coursetaking in the algebra-calculus pipeline on the likelihood of placing out of postsecondary remedial mathematics. A nonparametric variant of propensity score analysis was used on a nationally representative data set to remove selection bias and test for an effect…

The Profit Motive: The Bane of Mathematics Education.

ERIC Educational Resources Information Center

Koblitz, Neal

1992-01-01

Discusses the lack of calculus textbook improvements even though there have been complaints about them from students and professors. Argues against using commercial textbooks in calculus instruction. (ASK)

The AP Calculus Exam Reading Experience: Implications for Teacher Classroom Practice and Student Comprehension

ERIC Educational Resources Information Center

Corcoran, Mimi

2017-01-01

This dissertation explores the views and experiences of high school calculus teachers and college mathematics professors on the professional development which occurs at the annual national AP Calculus exam grading. This professional development experience comes in several forms: the exam briefing sessions, the actual reading of the exams, the…

Coordinating Multiple Representations in a Reform Calculus Textbook

ERIC Educational Resources Information Center

Chang, Briana L.; Cromley, Jennifer G.; Tran, Nhi

2015-01-01

Coordination of multiple representations (CMR) is widely recognized as a critical skill in mathematics and is frequently demanded in reform calculus textbooks. However, little is known about the prevalence of coordination tasks in such textbooks. We coded 707 instances of CMR in a widely used reform calculus textbook and analyzed the distributions…

Calculus Challenges: An Active Learning Approach

ERIC Educational Resources Information Center

Crawford, Pam; Moseley, Daniel; Nancarrow, Mike; Ward, Erika

2018-01-01

One of the greatest challenges facing students new to calculus is the ability to persevere in the face of failure. Whether the student is choosing an integration technique or a series test, calculus is often the first course in mathematics where the path to the solution is not prescribed in an algorithmic way. At Jacksonville University we…

The Association of Precollege Use of Calculators with Student Performance in College Calculus

ERIC Educational Resources Information Center

Mao, Yi; White, Tyreke; Sadler, Philip M.; Sonnert, Gerhard

2017-01-01

This study investigates how the use of calculators during high school mathematics courses is associated with student performance in introductory college calculus courses in the USA. Data were drawn from a nationally representative sample of 7087 students enrolled in college calculus at 134 colleges and universities. They included information about…

Calculus: An Active Approach with Projects.

ERIC Educational Resources Information Center

Hilbert, Steve; And Others

Ithaca College, in New York, has developed and tested a projects-based first-year calculus course over the last 3 years which uses the graphs of functions and physical phenomena to illustrate and motivate the major concepts of calculus and to introduce students to mathematical modeling. The course curriculum is designed to: (1) emphasize on the…

Student Created Calculus Movies Using Computers and the TI-92.

ERIC Educational Resources Information Center

Sher, Lawrence; Wilkinson, Patricia

The Mathematics Department at Borough of Manhattan Community College (BMCC) (New York) has been actively involved since 1988 in a serious and successful program to improve instruction, understanding, and retention for women and minority students in calculus courses. One result of this work has been students creating calculus animations using…

Educating about Sustainability while Enhancing Calculus

ERIC Educational Resources Information Center

Pfaff, Thomas J.

2011-01-01

We give an overview of why it is important to include sustainability in mathematics classes and provide specific examples of how to do this for a calculus class. We illustrate that when students use "Excel" to fit curves to real data, fundamentally important questions about sustainability become calculus questions about those curves. (Contains 5…

Commentary on A General Curriculum in Mathematics for Colleges.

ERIC Educational Resources Information Center

Committee on the Undergraduate Program in Mathematics, Berkeley, CA.

This document constitutes a complete revision of the report of the same name first published in 1965. A new list of basic courses is described, consisting of Calculus I, Calculus II, Elementary Linear Algebra, Multivariable Calculus I, Linear Algebra, and Introductory Modern Algebra. Commentaries outline the content and spirit of these courses in…

Coordinating Multiple Representations in a Reform Calculus Textbook

ERIC Educational Resources Information Center

Chang, Briana L.; Cromley, Jennifer G.; Tran, Nhi

2016-01-01

Coordination of multiple representations (CMR) is widely recognized as a critical skill in mathematics and is frequently demanded in reform calculus textbooks. However, little is known about the prevalence of coordination tasks in such textbooks. We coded 707 instances of CMR in a widely used reform calculus textbook and analyzed the distributions…

TIMSS Advanced 2015 and Advanced Placement Calculus & Physics. A Framework Analysis. Research in Review 2016-1

ERIC Educational Resources Information Center

Lazzaro, Christopher; Jones, Lee; Webb, David C.; Grover, Ryan; Di Giacomo, F. Tony; Marino, Katherine Adele

2016-01-01

This report will determine to what degree the AP Physics 1 and 2 and AP Calculus AB and BC frameworks are aligned with the Trends in International Mathematics and Science Study (TIMSS) Advanced Physics and Mathematics frameworks. This will enable an exploration of any differences in content coverage and levels of complexity, and will set the stage…

A Mathematics Software Database Update.

ERIC Educational Resources Information Center

Cunningham, R. S.; Smith, David A.

1987-01-01

Contains an update of an earlier listing of software for mathematics instruction at the college level. Topics are: advanced mathematics, algebra, calculus, differential equations, discrete mathematics, equation solving, general mathematics, geometry, linear and matrix algebra, logic, statistics and probability, and trigonometry. (PK)

Improving Student Success in Calculus I Using a Co-Requisite Calculus I Lab

ERIC Educational Resources Information Center

Vestal, Sharon Schaffer; Brandenburger, Thomas; Furth, Alfred

2015-01-01

This paper describes how one university mathematics department was able to improve student success in Calculus I by requiring a co-requisite lab for certain groups of students. The groups of students required to take the co-requisite lab were identified by analyzing student data, including Math ACT scores, ACT Compass Trigonometry scores, and…

Foundation Mathematics for the Physical Sciences

NASA Astrophysics Data System (ADS)

Riley, K. F.; Hobson, M. P.

2011-03-01

1. Arithmetic and geometry; 2. Preliminary algebra; 3. Differential calculus; 4. Integral calculus; 5. Complex numbers and hyperbolic functions; 6. Series and limits; 7. Partial differentiation; 8. Multiple integrals; 9. Vector algebra; 10. Matrices and vector spaces; 11. Vector calculus; 12. Line, surface and volume integrals; 13. Laplace transforms; 14. Ordinary differential equations; 15. Elementary probability; Appendices; Index.

Studies in Mathematics, Volume XV. Calculus and Science.

ERIC Educational Resources Information Center

Twersky, Victor

This book is designed to illustrate how one general method of calculus is used in many different sciences and how different methods of calculus have furthered the development of essentially one field of science. The material is written so that it could serve as a math-science supplement for many courses. Chapters included are: (1) Introduction;…

Student Solution Manual for Foundation Mathematics for the Physical Sciences

NASA Astrophysics Data System (ADS)

Riley, K. F.; Hobson, M. P.

2011-03-01

1. Arithmetic and geometry; 2. Preliminary algebra; 3. Differential calculus; 4. Integral calculus; 5. Complex numbers and hyperbolic functions; 6. Series and limits; 7. Partial differentiation; 8. Multiple integrals; 9. Vector algebra; 10. Matrices and vector spaces; 11. Vector calculus; 12. Line, surface and volume integrals; 13. Laplace transforms; 14. Ordinary differential equations; 15. Elementary probability; Appendix.

Teaching Children How to Include the Inversion Principle in Their Reasoning about Quantitative Relations

ERIC Educational Resources Information Center

Nunes, Terezinha; Bryant, Peter; Evans, Deborah; Bell, Daniel; Barros, Rossana

2012-01-01

The basis of this intervention study is a distinction between numerical calculus and relational calculus. The former refers to numerical calculations and the latter to the analysis of the quantitative relations in mathematical problems. The inverse relation between addition and subtraction is relevant to both kinds of calculus, but so far research…

Mathematics Underground

ERIC Educational Resources Information Center

Luther, Kenneth H.

2012-01-01

Mathematical modeling of groundwater flow is a topic at the intersection of mathematics and geohydrology and is rarely encountered in undergraduate mathematics. However, this subject is full of interesting and meaningful examples of truly "applied" mathematics accessible to undergraduates, from the pre-calculus to advanced mathematics levels. This…

An Evaluative Calculus Project: Applying Bloom's Taxonomy to the Calculus Classroom

ERIC Educational Resources Information Center

Karaali, Gizem

2011-01-01

In education theory, Bloom's taxonomy is a well-known paradigm to describe domains of learning and levels of competency. In this article I propose a calculus capstone project that is meant to utilize the sixth and arguably the highest level in the cognitive domain, according to Bloom et al.: evaluation. Although one may assume that mathematics is…

Calculus, Part 3, Student's Text, Unit No. 70. Revised Edition.

ERIC Educational Resources Information Center

Beck, A.; And Others

This is part three of a three-part SMSG calculus text for high school students. One of the goals of the text is to present calculus as a mathematical discipline as well as presenting its practical uses. The authors emphasize the importance of being able to interpret the concepts and theory in terms of models to which they apply. The text…

Teacher's Guide to Secondary Mathematics.

ERIC Educational Resources Information Center

Duval County Schools, Jacksonville, FL.

This is a teacher's guide to secondary school mathematics. Developed for use in the Duval County Public Schools, Jacksonville, Florida. Areas of mathematics covered are algebra, analysis, calculus, computer literacy, computer science, geometry, analytic geometry, general mathematics, consumer mathematics, pre-algebra, probability and statistics,…

Bunny hops: using multiplicities of zeroes in calculus for graphing

NASA Astrophysics Data System (ADS)

Miller, David; Deshler, Jessica M.; Hansen, Ryan

2016-07-01

Students learn a lot of material in each mathematics course they take. However, they are not always able to make meaningful connections between content in successive mathematics courses. This paper reports on a technique to address a common topic in calculus I courses (intervals of increase/decrease and concave up/down) while also making use of students' pre-existing knowledge about the behaviour of functions around zeroes based on multiplicities.

Calculation of spontaneous emission from a V-type three-level atom in photonic crystals using fractional calculus

DOE Office of Scientific and Technical Information (OSTI.GOV)

Huang, Chih-Hsien; Hsieh, Wen-Feng; Institute of Electro-Optical Science and Engineering, National Cheng Kung University, 1 Dahsueh Rd., Tainan 701, Taiwan

2011-07-15

Fractional time derivative, an abstract mathematical operator of fractional calculus, is used to describe the real optical system of a V-type three-level atom embedded in a photonic crystal. A fractional kinetic equation governing the dynamics of the spontaneous emission from this optical system is obtained as a fractional Langevin equation. Solving this fractional kinetic equation by fractional calculus leads to the analytical solutions expressed in terms of fractional exponential functions. The accuracy of the obtained solutions is verified through reducing the system into the special cases whose results are consistent with the experimental observation. With accurate physical results and avoidingmore » the complex integration for solving this optical system, we propose fractional calculus with fractional time derivative as a better mathematical method to study spontaneous emission dynamics from the optical system with non-Markovian dynamics.« less

The role of a posteriori mathematics in physics

NASA Astrophysics Data System (ADS)

MacKinnon, Edward

2018-05-01

The calculus that co-evolved with classical mechanics relied on definitions of functions and differentials that accommodated physical intuitions. In the early nineteenth century mathematicians began the rigorous reformulation of calculus and eventually succeeded in putting almost all of mathematics on a set-theoretic foundation. Physicists traditionally ignore this rigorous mathematics. Physicists often rely on a posteriori math, a practice of using physical considerations to determine mathematical formulations. This is illustrated by examples from classical and quantum physics. A justification of such practice stems from a consideration of the role of phenomenological theories in classical physics and effective theories in contemporary physics. This relates to the larger question of how physical theories should be interpreted.

Oliver Heaviside's "Dinner"

NASA Astrophysics Data System (ADS)

Giorello, Giulio; Sinigaglia, Corrado

In the following pages we begin, in the first chapter, with a reappraisal of some ideas of Edouard Le Roy about mathematical experience, mainly in relation with the history of complex numbers. In the second chapter we discuss in some detail the i-story, and we draw a comparison between "Imaginary Quantity" and Operational Calculus from the perspective of Heaviside's conceptions of the growth of mathematics. In the third chapter we reconstruct the δ-story, i.e. the Heaviside calculus leading to the constitution of a new mathematical object, the so-called Dirac's δ-function. Finally, in the last chapter, we bring together methodological and historical considerations in order to support Lakatos' idea of quasi-empiricism in mathematics.

Representations in Calculus: Two Contrasting Cases.

ERIC Educational Resources Information Center

Aspinwall, Leslie; Shaw, Kenneth L.

2002-01-01

Illustrates the contrasting thinking processes of two beginning calculus students' geometric and analytic schemes for the derivative function. Suggests that teachers can enhance students' understanding by continuing to demonstrate how different representations of the same mathematical concept provide additional information. (KHR)

Analysis of Errors and Misconceptions in the Learning of Calculus by Undergraduate Students

ERIC Educational Resources Information Center

Muzangwa, Jonatan; Chifamba, Peter

2012-01-01

This paper is going to analyse errors and misconceptions in an undergraduate course in Calculus. The study will be based on a group of 10 BEd. Mathematics students at Great Zimbabwe University. Data is gathered through use of two exercises on Calculus 1&2.The analysis of the results from the tests showed that a majority of the errors were due…

Using an Advanced Graphing Calculator in the Teaching and Learning of Calculus

ERIC Educational Resources Information Center

Leng, Ng Wee

2011-01-01

The purpose of this study was to investigate how the use of TI-Nspire[TM] could enhance the teaching and learning of calculus. A conceptual framework for the use of TI-Nspire[TM] for learning calculus in a mathematics classroom is proposed that describes the interactions among the students, TI-Nspire[TM], and the learning tasks, and how they lead…

Collaborate and Innovate: One Department's Perspective on Factors Supporting and Sustaining Pedagogical Change in Calculus I

ERIC Educational Resources Information Center

Schroeder, Larissa Bucchi; McGivney-Burelle, Jean; Haruta, Mako E.; Xue, Fei

2018-01-01

At the University of Hartford we transformed our approach to Calculus I--moving it away from a lecture-dominant format to one that focuses squarely on students solving problems and discussing and presenting their mathematical ideas for the majority of class time. In this article, we discuss our Flipping Calculus project and how a departmental…

The development and nature of problem-solving among first-semester calculus students

NASA Astrophysics Data System (ADS)

Dawkins, Paul Christian; Mendoza Epperson, James A.

2014-08-01

This study investigates interactions between calculus learning and problem-solving in the context of two first-semester undergraduate calculus courses in the USA. We assessed students' problem-solving abilities in a common US calculus course design that included traditional lecture and assessment with problem-solving-oriented labs. We investigate this blended instruction as a local representative of the US calculus reform movements that helped foster it. These reform movements tended to emphasize problem-solving as well as multiple mathematical registers and quantitative modelling. Our statistical analysis reveals the influence of the blended traditional/reform calculus instruction on students' ability to solve calculus-related, non-routine problems through repeated measures over the semester. The calculus instruction in this study significantly improved students' performance on non-routine problems, though performance improved more regarding strategies and accuracy than it did for drawing conclusions and providing justifications. We identified problem-solving behaviours that characterized top performance or attrition in the course. Top-performing students displayed greater algebraic proficiency, calculus skills, and more general heuristics than their peers, but overused algebraic techniques even when they proved cumbersome or inappropriate. Students who subsequently withdrew from calculus often lacked algebraic fluency and understanding of the graphical register. The majority of participants, when given a choice, relied upon less sophisticated trial-and-error approaches in the numerical register and rarely used the graphical register, contrary to the goals of US calculus reform. We provide explanations for these patterns in students' problem-solving performance in view of both their preparation for university calculus and the courses' assessment structure, which preferentially rewarded algebraic reasoning. While instruction improved students' problem-solving performance, we observe that current instruction requires ongoing refinement to help students develop multi-register fluency and the ability to model quantitatively, as is called for in current US standards for mathematical instruction.

Calculus Students' Understanding of Volume

ERIC Educational Resources Information Center

Dorko, Allison; Speer, Natasha M.

2013-01-01

Researchers have documented difficulties that elementary school students have in understanding volume. Despite its importance in higher mathematics, we know little about college students' understanding of volume. This study investigated calculus students' understanding of volume. Clinical interview transcripts and written responses to volume…

Secondary Schools Curriculum Guide, Mathematics, Grades 10-12. Revised.

ERIC Educational Resources Information Center

Cranston School Dept., RI.

Behavioral objectives for grades 10 through 12 are specified for plane geometry, algebra, general mathematics, computer mathematics, slide rule mathematics, basic college mathematics, trigonometry, analytic geometry, calculus and probability. Most sections present material in terms of portions of a school year. At least one major objective is…

Mathematics for Physics

NASA Astrophysics Data System (ADS)

Stone, Michael; Goldbart, Paul

2009-07-01

Preface; 1. Calculus of variations; 2. Function spaces; 3. Linear ordinary differential equations; 4. Linear differential operators; 5. Green functions; 6. Partial differential equations; 7. The mathematics of real waves; 8. Special functions; 9. Integral equations; 10. Vectors and tensors; 11. Differential calculus on manifolds; 12. Integration on manifolds; 13. An introduction to differential topology; 14. Group and group representations; 15. Lie groups; 16. The geometry of fibre bundles; 17. Complex analysis I; 18. Applications of complex variables; 19. Special functions and complex variables; Appendixes; Reference; Index.

The Vector Calculus Gap: Mathematics (Does Not Equal) Physics.

ERIC Educational Resources Information Center

Dray, Tevian; Manogue, Corinne A.

1999-01-01

Discusses some of the differences between the ways mathematicians and physicists view vector calculus and the gap between the way this material is traditionally taught by mathematicians and the way physicists use it. Suggests some ways to narrow the gap. (Author/ASK)

Calculus and Sailing.

ERIC Educational Resources Information Center

Palmaccio, Richard J.

1982-01-01

A method of using vector analysis is presented that is an application of calculus that helps to find the best angle for tacking a boat into the wind. While the discussion is theoretical, it is seen as a good illustration of mathematical investigation of a given situation. (MP)

The Case for Biocalculus: Design, Retention, and Student Performance.

PubMed

Eaton, Carrie Diaz; Highlander, Hannah Callender

2017-01-01

Calculus is one of the primary avenues for initial quantitative training of students in all science, technology, engineering, and mathematics fields, but life science students have been found to underperform in the traditional calculus setting. As a result, and because of perceived lack of its contribution to the understanding of biology, calculus is being actively cut from biology program requirements at many institutions. Here, we present an alternative: a model for learning mathematics that sees the partner disciplines as crucial to student success. We equip faculty with information to engage in dialogue within and between disciplinary departments involved in quantitative education. This includes presenting a process for interdisciplinary development and implementation of biology-oriented Calculus I courses at two institutions with different constituents, goals, and curricular constraints. When life science students enrolled in these redesigned calculus courses are compared with life science students enrolled in traditional calculus courses, students in the redesigned calculus courses learn calculus concepts and skills as well as their traditional course peers; however, the students in the redesigned courses experience more authentic life science applications and are more likely to stay and succeed in the course than their peers who are enrolled in traditional courses. Therefore, these redesigned calculus courses hold promise in helping life science undergraduate students attain Vision and Change recommended competencies. © 2017 C. D. Eaton and H. C. Highlander. CBE—Life Sciences Education © 2017 The American Society for Cell Biology. This article is distributed by The American Society for Cell Biology under license from the author(s). It is available to the public under an Attribution–Noncommercial–Share Alike 3.0 Unported Creative Commons License (http://creativecommons.org/licenses/by-nc-sa/3.0).

Preparing Future College Instructors: The Role of Graduate Student Teaching Assistants (GTAs) in Successful College Calculus Programs

NASA Astrophysics Data System (ADS)

Ellis, Jessica Fabricant

Graduate student Teaching Assistants (GTAs) contribute to calculus instruction in two ways: as the primary teacher and as recitation leaders. GTAs can also be viewed as the next generation of mathematics instructors. Thus, in addition to their immediate contribution to the landscape of Calculus 1 instruction, GTAs will contribute significantly to the long-term state of calculus in their future occupations. However, their preparation for these roles varies widely and is often minimal. In this study, I first compare the mathematical beliefs, instructional practices, and student success of GTAs to other Calculus 1 instructors. I then provide rich descriptions for three GTA professional development (PD) programs that prepare graduate students as course instructors, as recitation leaders, and as future faculty. I then investigate the instructional practices and mathematical beliefs of graduate students coming from these three PD programs. I conclude this work with a description of a framework for GTA-PD programs. To accomplish this work, I conducted a mixed-method analysis on national survey data and case study data from four doctoral granting institutions. These four institutions were chosen because of their higher-than-expected student success in Calculus 1. The results of these analyses indicate that graduate students teach in more innovative ways than other instructors, though their students were less successful. Among the four case study institutions, I identified three models of GTA-PD, each of which appeared successful in accomplishing their goals. These goals included transitioning graduate students into the role of instructor, preparing graduate students to implement an innovative approach to Calculus 1, and supporting graduate students as recitation leaders. These analyses also led to the development of a framework to be used to characterize, evaluate, and consider the implementation of graduate student professional development programs. This GTA-PD framework is thus one of the major contributions put forth by this dissertation.

Total Quality Management in the Classroom: Applications to University-Level Mathematics.

ERIC Educational Resources Information Center

Williams, Frank

1995-01-01

Describes a Total Quality Management-based system of instruction that is used in a variety of undergraduate mathematics courses. The courses that incorporate this approach include mathematics appreciation, introductory calculus, and advanced applied linear algebra. (DDR)

Investigations in Mathematics Education, Vol. 10, No. 4.

ERIC Educational Resources Information Center

Osborne, Alan R., Ed.

Eighteen research reports related to mathematics education are abstracted and analyzed. Four of the reports deal with aspects of learning theory, five with topics in mathematics instruction (history of mathematics, exponents, probability, calculus, and calculators), four with teacher characteristics, and one each with testing, student interests,…

"The Age of Newton": An Intensive Physics and Mathematics Course

ERIC Educational Resources Information Center

Calvert, J. B.; And Others

1976-01-01

Describes an intensive course in mathematics (calculus), mechanics, optics, and astronomy directed mainly toward nonscience students. Course format, operation, and student evaluation appear. (Author/CP)

Computer Activities for College Algebra and Precalculus.

ERIC Educational Resources Information Center

White, Jacci Wozniak; Norwich, Vicki Howard

Mathematics software can be a great aid in understanding difficult mathematics concepts at all levels. This paper presents nine exercises on calculus concepts by using different software used in mathematics education. Each exercise includes instruction on how to use software in order to highlight a specific concept in mathematics. This paper also…

A New Start for Mathematics Curriculum.

ERIC Educational Resources Information Center

Tucker, Alan

Arguing that a major re-thinking of the mathematics curriculum is needed, this paper urges two-year colleges to take the lead in curriculum revision. Section I suggests that the pre-calculus orientation of high school mathematics may be inappropriate, viewing mathematics related to computers and dependent on computers for computation as more…

Using Technology to Promote Mathematical Discourse Concerning Women in Mathematics

ERIC Educational Resources Information Center

Phy, Lyn

2008-01-01

This paper discusses uses of technology to facilitate mathematical discourse concerning women in mathematics. Such a topic can be introduced in various traditional courses such as algebra, geometry, trigonometry, probability and statistics, or calculus, but it is not included in traditional textbooks. Through the ideas presented here, you can…

Students Build Mathematical Theory: Semantic Warrants in Argumentation

ERIC Educational Resources Information Center

Walter, Janet G.; Barros, Tara

2011-01-01

In this paper, we explore the development of two grounded theories. One theory is mathematical and grounded in the work of university calculus students' collaborative development of mathematical methods for finding the volume of a solid of revolution, in response to mathematical necessity in problem solving, without prior instruction on solution…

University Students' Problem Posing Abilities and Attitudes towards Mathematics.

ERIC Educational Resources Information Center

Grundmeier, Todd A.

2002-01-01

Explores the problem posing abilities and attitudes towards mathematics of students in a university pre-calculus class and a university mathematical proof class. Reports a significant difference in numeric posing versus non-numeric posing ability in both classes. (Author/MM)

Using CAS to Solve Classical Mathematics Problems

ERIC Educational Resources Information Center

Burke, Maurice J.; Burroughs, Elizabeth A.

2009-01-01

Historically, calculus has displaced many algebraic methods for solving classical problems. This article illustrates an algebraic method for finding the zeros of polynomial functions that is closely related to Newton's method (devised in 1669, published in 1711), which is encountered in calculus. By exploring this problem, precalculus students…

On Flipping First-Semester Calculus: A Case Study

ERIC Educational Resources Information Center

Petrillo, Joseph

2016-01-01

High failure rates in calculus have plagued students, teachers, and administrators for decades, while science, technology, engineering, and mathematics programmes continue to suffer from low enrollments and high attrition. In an effort to affect this reality, some educators are "flipping" (or inverting) their classrooms. By flipping, we…

Making Implicit Multivariable Calculus Representations Explicit: A Clinical Study

ERIC Educational Resources Information Center

McGee, Daniel; Moore-Russo, Deborah; Martinez-Planell, Rafael

2015-01-01

Reviewing numerous textbooks, we found that in both differential and integral calculus textbooks the authors commonly assume that: (i) students can generalize associations between representations in two dimensions to associations between representations of the same mathematical concept in three dimensions on their own; and (ii) explicit…

Computerized Business Calculus Using Calculators, Examples from Mathematics to Finance.

ERIC Educational Resources Information Center

Vest, Floyd

1991-01-01

After discussing the role of supercalculators within the business calculus curriculum, several examples are presented which allow the reader to examine the capabilities and codes of calculators specific to different major manufacturers. The topics examined include annuities, Newton's method, fixed point iteration, graphing, solvers, and…

Line integral on engineering mathematics

NASA Astrophysics Data System (ADS)

Wiryanto, L. H.

2018-01-01

Definite integral is a basic material in studying mathematics. At the level of calculus, calculating of definite integral is based on fundamental theorem of calculus, related to anti-derivative, as the inverse operation of derivative. At the higher level such as engineering mathematics, the definite integral is used as one of the calculating tools of line integral. the purpose of this is to identify if there is a question related to line integral, we can use definite integral as one of the calculating experience. The conclusion of this research says that the teaching experience in introducing the relation between both integrals through the engineer way of thinking can motivate and improve students in understanding the material.

The Use of Applets for Developing Understanding in Mathematics: A Case Study Using Maplets for Calculus with Continuity Concepts

ERIC Educational Resources Information Center

Patenaude, Raymond E.

2013-01-01

The Common Core State Standards for Mathematics (CCSSM) are founded on a long history of mathematics education research emphasizing the importance of teaching mathematics for understanding. The CCSSM along with the National Council of Teachers of Mathematics (NCTM) recommend the use of technology in the teaching of mathematics. New mobile…

Hermeneutics of differential calculus in eighteenth-century northern Germany.

PubMed

Blanco, Mónica

2008-01-01

This paper applies comparative textbook analysis to studying the mathematical development of differential calculus in northern German states during the eighteenth century. It begins with describing how the four textbooks analyzed presented the foundations of calculus and continues with assessing the influence each of these foundational approaches exerted on the resolution of problems, such as the determination of tangents and extreme values, and even on the choice of coordinates for both algebraic and transcendental curves.

Difficulties First-Year University Mathematics Students Have in Reading Their Mathematics Textbook. Technical Report. No. 2009-1

ERIC Educational Resources Information Center

Shepherd, Mary D.; Selden, Annie; Selden, John

2009-01-01

This exploratory study examined the experiences and difficulties certain first-year university students displayed in reading new passages from their mathematics textbooks. We interviewed eleven precalculus and calculus students who were considered to be good at mathematics, as indicated by high ACT mathematics scores. These students were also …

Mathematical thinking and origami

NASA Astrophysics Data System (ADS)

Wares, Arsalan

2016-01-01

The purpose of this paper is to describe the mathematics that emanates from the construction of an origami box. We first construct a simple origami box from a rectangular sheet and then discuss some of the mathematical questions that arise in the context of geometry and calculus.

Predicting Performance in a First Engineering Calculus Course: Implications for Interventions

ERIC Educational Resources Information Center

Hieb, Jeffrey L.; Lyle, Keith B.; Ralston, Patricia A. S.; Chariker, Julia

2015-01-01

At the University of Louisville, a large, urban institution in the south-east United States, undergraduate engineering students take their mathematics courses from the school of engineering. In the fall of their freshman year, engineering students take "Engineering Analysis I," a calculus-based engineering analysis course. After the…

The Characteristic of the Process of Students' Metacognition in Solving Calculus Problems

ERIC Educational Resources Information Center

Purnomo, Dwi; Nusantara, Toto; Subanji; Rahardjo, Swasono

2017-01-01

This article is the result of research aims to describe the patterns and characteristics of the process of metacognition student of mathematics in solving calculus problems. Description was done by looking at changes in "awareness," "evaluation," and "regulation" as components of metacognition. The changes in…

Some Problems of Extremes in Geometry and Construction

ERIC Educational Resources Information Center

Yanovsky, Levi

2008-01-01

Two original problems in geometry are presented with solutions utilizing to differential calculus: (a) rectangle inscribed in a sector; (b) point on the ray of the angle. The possibility of applying mathematics in general and differential calculus in particular for solution of practical problems is discussed. (Contains 8 figures.)

A Graphical Introduction to the Derivative

ERIC Educational Resources Information Center

Samuels, Jason

2017-01-01

Calculus has frequently been called one the greatest intellectual achievements of humankind. As a key transitional course to college mathematics, it combines such elementary ideas as rate with new abstract ideas--such as infinity, instantaneous change, and limit--to formulate the derivative and the integral. Most calculus texts begin with the…

Simplifying the Mathematical Treatment of Radioactive Decay

ERIC Educational Resources Information Center

Auty, Geoff

2011-01-01

Derivation of the law of radioactive decay is considered without prior knowledge of calculus or the exponential series. Calculus notation and exponential functions are used because ultimately they cannot be avoided, but they are introduced in a simple way and explained as needed. (Contains 10 figures, 1 box, and 1 table.)

Calculus in Elementary School: An Example of ICT-Based Curriculum Transformation

ERIC Educational Resources Information Center

Fluck, Andrew; Ranmuthugala, Dev; Chin, Chris; Penesis, Irene

2012-01-01

Integral calculus is generally regarded as a fundamental but advanced aspect of mathematics, and it is not generally studied until students are aged about fifteen or older. Understanding the transformative potential of information and communication technology, this project undertook an investigation in four Australian schools to train students…

A Transformative Model for Undergraduate Quantitative Biology Education

ERIC Educational Resources Information Center

Usher, David C.; Driscoll, Tobin A.; Dhurjati, Prasad; Pelesko, John A.; Rossi, Louis F.; Schleiniger, Gilberto; Pusecker, Kathleen; White, Harold B.

2010-01-01

The "BIO2010" report recommended that students in the life sciences receive a more rigorous education in mathematics and physical sciences. The University of Delaware approached this problem by (1) developing a bio-calculus section of a standard calculus course, (2) embedding quantitative activities into existing biology courses, and (3)…

Design Research on Inquiry-Based Multivariable Calculus: Focusing on Students' Argumentation and Instructional Design

ERIC Educational Resources Information Center

Kwon, Oh Nam; Bae, Younggon; Oh, Kuk Hwan

2015-01-01

In this study, researchers design and implement an inquiry based multivariable calculus course in a university which aims at enhancing students' argumentation in rich mathematical discussions. This research aims to understand the characteristics of students' argumentation in activities involving proof constructions through mathematical…

Using Matlab in a Multivariable Calculus Course.

ERIC Educational Resources Information Center

Schlatter, Mark D.

The benefits of high-level mathematics packages such as Matlab include both a computer algebra system and the ability to provide students with concrete visual examples. This paper discusses how both capabilities of Matlab were used in a multivariate calculus class. Graphical user interfaces which display three-dimensional surfaces, contour plots,…

A Calculus Activity with Foundations in Geometric Learning

ERIC Educational Resources Information Center

Wagner, Jennifer; Sharp, Janet

2017-01-01

Calculus, perhaps more than other areas of mathematics, has a reputation for being steeped with procedures. In fact, through the years, it has been noticed of many students getting caught in the trap of trying to memorize algorithms and rules without developing associated concept knowledge. Specifically, students often struggle with the…

Writing-to-Learn Activities to Provoke Deeper Learning in Calculus

ERIC Educational Resources Information Center

Jaafar, Reem

2016-01-01

For students with little experience in mathematical thinking and conceptualization, writing-to-learn activities (WTL) can be particularly effective in promoting discovery and understanding. For community college students embarking on a first calculus course in particular, writing activities can help facilitate the transition from an "apply…

An Investigation of Calculus Learning Using Factorial Modeling.

ERIC Educational Resources Information Center

Dick, Thomas P.; Balomenos, Richard H.

Structural covariance models that would explain the correlations observed among mathematics achievement and participation measures and related cognitive and affective variables were developed. A sample of college calculus students (N=268; 124 females and 144 males) was administered a battery of cognitive tests (including measures of spatial-visual…

Encouraging Example Generation: A Teaching Experiment in First-Semester Calculus

ERIC Educational Resources Information Center

Wagner, Elaine Rumsey; Orme, Susan Marla; Turner, Heidi Jean; Yopp, David

2017-01-01

Mathematicians use example generation to test and verify mathematical ideas; however, the processes through which undergraduates learn to productively generate examples are not well understood. We engaged calculus students in a teaching experiment designed to develop skills in productively generating examples to learn novel concepts. This article…

Are Homeschoolers Prepared for College Calculus?

ERIC Educational Resources Information Center

Wilkens, Christian P.; Wade, Carol H.; Sonnert, Gerhard; Sadler, Philip M.

2015-01-01

Homeschooling in the United States has grown considerably over the past several decades. This article presents findings from the Factors Influencing College Success in Mathematics (FICSMath) survey, a national study of 10,492 students enrolled in tertiary calculus, including 190 students who reported homeschooling for a majority of their high…

Triple Play: From De Morgan to Stirling to Euler to Maclaurin to Stirling

ERIC Educational Resources Information Center

Kolpas, Sid

2011-01-01

Augustus De Morgan (1806-1871) was a significant Victorian Mathematician who made contributions to mathematics history, mathematical recreations, mathematical logic, calculus, and probability and statistics. He was an inspiring mathematics professor who influenced many of his students to join the profession. One of De Morgan's significant books…

Secondary School Mathematics Curriculum Improvement Study Information Bulletin 7.

ERIC Educational Resources Information Center

Secondary School Mathematics Curriculum Improvement Study, New York, NY.

The background, objectives, and design of Secondary School Mathematics Curriculum Improvement Study (SSMCIS) are summarized. Details are given of the content of the text series, "Unified Modern Mathematics," in the areas of algebra, geometry, linear algebra, probability and statistics, analysis (calculus), logic, and computer…

Unpacking the Logic of Mathematical Statements.

ERIC Educational Resources Information Center

Selden, John; Selden, Annie

1995-01-01

Investigated (n=61) undergraduates' ability to unpack informally written mathematical statements into the language of predicate calculus in an introduction to proofs and mathematical reasoning. Found that students were unable to construct proofs or validate them. Appendices are "A Sample Validation" and "Building a Statement Image." (MKR)

Integrating Mathematics into the Introductory Biology Laboratory Course

ERIC Educational Resources Information Center

White, James D.; Carpenter, Jenna P.

2008-01-01

Louisiana Tech University has an integrated science curriculum for its mathematics, chemistry, physics, computer science, biology-research track and secondary mathematics and science education majors. The curriculum focuses on the calculus sequence and introductory labs in biology, physics, and chemistry. In the introductory biology laboratory…

Cause-effect analysis: improvement of a first year engineering students' calculus teaching model

NASA Astrophysics Data System (ADS)

van der Hoff, Quay; Harding, Ansie

2017-01-01

This study focuses on the mathematics department at a South African university and in particular on teaching of calculus to first year engineering students. The paper reports on a cause-effect analysis, often used for business improvement. The cause-effect analysis indicates that there are many factors that impact on secondary school teaching of mathematics, factors that the tertiary sector has no control over. The analysis also indicates the undesirable issues that are at the root of impeding success in the calculus module. Most important is that students are not encouraged to become independent thinkers from an early age. This triggers problems in follow-up courses where students are expected to have learned to deal with the work load and understanding of certain concepts. A new model was designed to lessen the impact of these undesirable issues.

Mathematical Methods for Optical Physics and Engineering

NASA Astrophysics Data System (ADS)

Gbur, Gregory J.

2011-01-01

1. Vector algebra; 2. Vector calculus; 3. Vector calculus in curvilinear coordinate systems; 4. Matrices and linear algebra; 5. Advanced matrix techniques and tensors; 6. Distributions; 7. Infinite series; 8. Fourier series; 9. Complex analysis; 10. Advanced complex analysis; 11. Fourier transforms; 12. Other integral transforms; 13. Discrete transforms; 14. Ordinary differential equations; 15. Partial differential equations; 16. Bessel functions; 17. Legendre functions and spherical harmonics; 18. Orthogonal functions; 19. Green's functions; 20. The calculus of variations; 21. Asymptotic techniques; Appendices; References; Index.

Mathematical Features of the Calculus

ERIC Educational Resources Information Center

Sauerheber, Richard D.

2010-01-01

The fundamental theorems of the calculus describe the relationships between derivatives and integrals of functions. The value of any function at a particular location is the definite derivative of its integral and the definite integral of its derivative. Thus, any value is the magnitude of the slope of the tangent of its integral at that position,…

Constructivized Calculus in College Mathematics

ERIC Educational Resources Information Center

Lawrence, Barbara Ann

2012-01-01

The purpose of this study is to present some of the classical concepts, definitions, and theorems of calculus from the constructivists' point of view in the spirit of the philosophies of L.E.J. Brouwer and Errett Bishop. This presentation will compare the classical statements to the constructivized statements. The method focuses on giving…

Calculus Limits Involving Infinity: The Role of Students' Informal Dynamic Reasoning

ERIC Educational Resources Information Center

Jones, Steven R.

2015-01-01

Few studies on calculus limits have centred their focus on student understanding of limits at infinity or infinite limits that involve continuous functions (as opposed to discrete sequences). This study examines student understanding of these types of limits using both pure mathematics and applied-science functions and formulas. Seven calculus…

Understanding Introductory Students' Application of Integrals in Physics from Multiple Perspectives

ERIC Educational Resources Information Center

Hu, Dehui

2013-01-01

Calculus is used across many physics topics from introductory to upper-division level college courses. The concepts of differentiation and integration are important tools for solving real world problems. Using calculus or any mathematical tool in physics is much more complex than the straightforward application of the equations and algorithms that…

Assessing the Impact of Computer Programming in Understanding Limits and Derivatives in a Secondary Mathematics Classroom

ERIC Educational Resources Information Center

de Castro, Christopher H.

2011-01-01

This study explored the development of student's conceptual understandings of limit and derivative when utilizing specifically designed computational tools. Fourteen students from a secondary Advanced Placement Calculus AB course learned and explored the limit and derivative concepts from differential calculus using visualization tools in the…

Reference Framework for Describing and Assessing Students' Understanding in First Year Calculus

ERIC Educational Resources Information Center

Kannemeyer, Larry

2005-01-01

This paper presents aspects of a study that investigates the development of an instrument, a reference framework, to analyse students' written responses to non-routine problems in a first year calculus course in order to describe the complexities of their understanding and to assess their understanding of particular mathematical concepts.…

Calculus: Readings from the "Mathematics Teacher."

ERIC Educational Resources Information Center

Grinstein, Louise S., Ed.; Michaels, Brenda, Ed.

Many of the suggestions that calculus instructors have published as articles from 1908 through 1973 are included in this book of readings. The main criterion for selecting an item was whether it would be helpful to teachers and students; therefore, those which dealt exclusively with curricular structure were not included. The selected articles are…

Struggles and Successes Implementing Classroom Communication Technology in a College Pre-Calculus Course

ERIC Educational Resources Information Center

Case, Erin; Pape, Stephen

2013-01-01

This case study documents the struggles and successes encountered by a pre-calculus teacher while using Classroom Connectivity Technology (CCT) daily in her community college mathematics course. CCT refers to a wireless communication system that connects a teacher's computer with an individual student's handheld calculator and has been associated…

The Case for Biocalculus: Design, Retention, and Student Performance

ERIC Educational Resources Information Center

Eaton, Carrie Diaz; Highlander, Hannah Callender

2017-01-01

Calculus is one of the primary avenues for initial quantitative training of students in all science, technology, engineering, and mathematics fields, but life science students have been found to underperform in the traditional calculus setting. As a result, and because of perceived lack of its contribution to the understanding of biology, calculus…

A TENTATIVE GUIDE, DIFFERENTIAL AND INTEGRAL CALCULUS.

ERIC Educational Resources Information Center

BRANT, VINCENT; GERARDI, WILLIAM

THE COURSE IS INTENDED TO GO BEYOND THE REQUIREMENTS OF THE ADVANCED PLACEMENT PROGRAM IN MATHEMATICS AS DESIGNED BY THE COLLEGE ENTRANCE EXAMINATION BOARD. THE ADVANCED PLACEMENT PROGRAM CONSISTS OF A 1-YEAR COURSE COMBINING ANALYTIC GEOMETRY AND CALCULUS. PRESUPPOSED HERE ARE--A SEMESTER COURSE IN ANALYTIC GEOMETRY AND A THOROUGH KNOWLEDGE OF…

Teaching and Learning Calculus in Secondary Schools with the TI-Nspire

ERIC Educational Resources Information Center

Parrot, Mary Ann Serdina; Eu, Leong Kwan

2014-01-01

Technology can help develop understanding of abstract mathematical concepts through visualisation and graphic representation. The teaching and learning of calculus can be challenging as it involves abstract and complex ideas. The purpose of this study was to investigate how students and teachers attempt to use TI-Nspire, the latest graphing…

Contrasting Cases of Calculus Students' Understanding of Derivative Graphs

ERIC Educational Resources Information Center

Haciomeroglu, Erhan Selcuk; Aspinwall, Leslie; Presmeg, Norma C.

2010-01-01

This study adds momentum to the ongoing discussion clarifying the merits of visualization and analysis in mathematical thinking. Our goal was to gain understanding of three calculus students' mental processes and images used to create meaning for derivative graphs. We contrast the thinking processes of these three students as they attempted to…

Multivariate Limits and Continuity: A Survey of Calculus Textbooks.

ERIC Educational Resources Information Center

Thompson, Thomas M.; Wiggins, Kenneth L.

There has been much recent discussion concerning the content of the standard calculus course for students majoring in mathematics and the sciences. Some of this discussion has focused on the available textbooks. One weakness noted in some of these books involves the definitions of limit and continuity for functions of several variables. A…

Examining Student Agency in an Active-Learning Business Calculus Course

ERIC Educational Resources Information Center

Higgins, Abigail L.

2017-01-01

This study explored student agency in an active-learning business calculus course. The lecture-style instructional practices typically used in this course at this institution allow few opportunities for students to interact with their peers, interface with the instructor one-on-one, or do mathematics during class time. Additionally, this course…

A Lesson in Vectors "Plain" and Simple

ERIC Educational Resources Information Center

Bradshaw, David M.

2004-01-01

The United States Military Academy (USMA) has a four course core mathematics curriculum that is studied by all students. The third course is MA205, Calculus II; a multivariate calculus course filled with practical applications. During a Problem Solving Lab (PSL), students participated in a hands-on exercise with multiple vector operations,…

Connecting the Dots: Rediscovering Continuity

ERIC Educational Resources Information Center

Camenga, Kristin A.; Yates, Rebekah B. Johnson

2014-01-01

The topic of continuity is typically not introduced until calculus and then reexamined in real analysis. Recognizing the connections between secondary school mathematics and the advanced mathematics studied at the college level allows teachers to better identify mathematical concepts in student ideas, motivate students by piquing their curiosity,…

Mathematical Modelling in Engineering: A Proposal to Introduce Linear Algebra Concepts

ERIC Educational Resources Information Center

Cárcamo Bahamonde, Andrea; Gómez Urgelles, Joan; Fortuny Aymemí, Josep

2016-01-01

The modern dynamic world requires that basic science courses for engineering, including linear algebra, emphasise the development of mathematical abilities primarily associated with modelling and interpreting, which are not exclusively calculus abilities. Considering this, an instructional design was created based on mathematical modelling and…

Formalization of the Integral Calculus in the PVS Theorem Prover

NASA Technical Reports Server (NTRS)

Butler, Ricky W.

2004-01-01

The PVS Theorem prover is a widely used formal verification tool used for the analysis of safety-critical systems. The PVS prover, though fully equipped to support deduction in a very general logic framework, namely higher-order logic, it must nevertheless, be augmented with the definitions and associated theorems for every branch of mathematics and Computer Science that is used in a verification. This is a formidable task, ultimately requiring the contributions of researchers and developers all over the world. This paper reports on the formalization of the integral calculus in the PVS theorem prover. All of the basic definitions and theorems covered in a first course on integral calculus have been completed.The theory and proofs were based on Rosenlicht's classic text on real analysis and follow the traditional epsilon-delta method. The goal of this work was to provide a practical set of PVS theories that could be used for verification of hybrid systems that arise in air traffic management systems and other aerospace applications. All of the basic linearity, integrability, boundedness, and continuity properties of the integral calculus were proved. The work culminated in the proof of the Fundamental Theorem Of Calculus. There is a brief discussion about why mechanically checked proofs are so much longer than standard mathematics textbook proofs.

Teaching Mathematics to Civil Engineers

ERIC Educational Resources Information Center

Sharp, J. J.; Moore, E.

1977-01-01

This paper outlines a technique for teaching a rigorous course in calculus and differential equations which stresses applicability of the mathematics to problems in civil engineering. The method involves integration of subject matter and team teaching. (SD)

Intitialization, Conceptualization, and Application in the Generalized Fractional Calculus

NASA Technical Reports Server (NTRS)

Lorenzo, Carl F.; Hartley, Tom T.

1998-01-01

This paper provides a formalized basis for initialization in the fractional calculus. The intent is to make the fractional calculus readily accessible to engineering and the sciences. A modified set of definitions for the fractional calculus is provided which formally include the effects of initialization. Conceptualizations of fractional derivatives and integrals are shown. Physical examples of the basic elements from electronics are presented along with examples from dynamics, material science, viscoelasticity, filtering, instrumentation, and electrochemistry to indicate the broad application of the theory and to demonstrate the use of the mathematics. The fundamental criteria for a generalized calculus established by Ross (1974) are shown to hold for the generalized fractional calculus under appropriate conditions. A new generalized form for the Laplace transform of the generalized differintegral is derived. The concept of a variable structure (order) differintegral is presented along with initial efforts toward meaningful definitions.

Initialization, conceptualization, and application in the generalized (fractional) calculus.

PubMed

Lorenzo, Carl F; Hartley, Tom T

2007-01-01

This paper provides a formalized basis for initialization in the fractional calculus. The intent is to make the fractional calculus readily accessible to engineering and the sciences. A modified set of definitions for the fractional calculus is provided which formally include the effects of initialization. Conceptualizations of fractional derivatives and integrals are shown. Physical examples of the basic elements from electronics are presented along with examples from dynamics, material science, viscoelasticity, filtering, instrumentation, and electrochemistry to indicate the broad application of the theory and to demonstrate the use of the mathematics. The fundamental criteria for a generalized calculus established by Ross (1974) are shown to hold for the generalized fractional calculus under appropriate conditions. A new generalized form for the Laplace transform of the generalized differintegral is derived. The concept of a variable structure (order) differintegral is presented along with initial efforts toward meaningful definitions.

Improving Student Success in Calculus: A Comparison of Four College Calculus Classes

NASA Astrophysics Data System (ADS)

Bagley, Spencer Franklin

The quality of education in science, technology, engineering, and mathematics (STEM) fields is an issue of particular educational and economic importance, and Calculus I is a linchpin course in STEM major tracks. A national study is currently being conducted examining the characteristics of successful programs in college calculus (CSPCC, 2012). In work related to the CSPCC program, this study examines the effects on student outcomes of four different teaching strategies used at a single institution. The four classes were a traditional lecture, a lecture with discussion, a lecture incorporating both discussion and technology, and an inverted model. This dissertation was guided by three questions: (1) What impact do these four instructional approaches have on students' persistence, beliefs about mathematics, and conceptual and procedural achievement in calculus? (2) How do students at the local institution compare to students in the national database? And (3) How do the similarities and differences in opportunities for learning presented in the four classes contribute to the similarities and differences in student outcomes? Quantitative analysis of surveys and exams revealed few statistically significant differences in outcomes, and students in the inverted classroom often had poorer outcomes than those in other classes. Students in the technology-enhanced class scored higher on conceptual items on the final exam than those in other classes. Comparing to the national database, local students had similar switching rates but less expert-like attitudes and beliefs about mathematics than the national average. Qualitative analysis of focus group interviews, classroom observations, and student course evaluations showed that several implementation issues, some the result of pragmatic constraints, others the result of design choice, weakened affordances provided by innovative features and shrunk the differences between classes. There were substantial differences between the inverted classroom in this study and successful implementations in the literature. I identified a set of departures that forms a list of best practices for inverting classrooms. Students in all classes felt that prior calculus experience was a prerequisite for their current calculus class, and that class sessions felt rushed. These concerns implicate the constraints imposed by the curriculum shared by the four classes.

The Vector Space as a Unifying Concept in School Mathematics.

ERIC Educational Resources Information Center

Riggle, Timothy Andrew

The purpose of this study was to show how the concept of vector space can serve as a unifying thread for mathematics programs--elementary school to pre-calculus college level mathematics. Indicated are a number of opportunities to demonstrate how emphasis upon the vector space structure can enhance the organization of the mathematics curriculum.…

PREFACE: Fractional Differentiation and its Applications (FDA08) Fractional Differentiation and its Applications (FDA08)

NASA Astrophysics Data System (ADS)

Baleanu, Dumitru; Tenreiro Machado, J. A.

2009-10-01

The international workshop, Fractional Differentiation and its Applications (FDA08), held at Cankaya University, Ankara, Turkey on 5-7 November 2008, was the third in an ongoing series of conferences dedicated to exploring applications of fractional calculus in science, engineering, economics and finance. Fractional calculus, which deals with derivatives and integrals of any order, is now recognized as playing an important role in modeling multi-scale problems that span a wide range of time or length scales. Fractional calculus provides a natural link to the intermediate-order dynamics that often reflects the complexity of micro- and nanostructures through fractional-order differential equations. Unlike the more established techniques of mathematical physics, the methods of fractional differentiation are still under development; while it is true that the ideas of fractional calculus are as old as the classical integer-order differential operators, modern work is proceeding by both expanding the capabilities of this mathematical tool and by widening its range of applications. Hence, the interested reader will find papers here that focus on the underlying mathematics of fractional calculus, that extend fractional-order operators into new domains, and that apply well established methods to experimental and theoretical problems. The organizing committee invited presentations from experts representing the international community of scholars in fractional calculus and welcomed contributions from the growing number of researchers who are applying fractional differentiation to complex technical problems. The selection of papers in this topical issue of Physica Scripta reflects the success of the FDA08 workshop, with the emergence of a variety of novel areas of application. With these ideas in mind, the guest editors would like to honor the many distinguished scientists that have promoted the development of fractional calculus and, in particular, Professor George M Zaslavsky who supported this special issue but passed away recently. The organizing committee wishes to thank the sponsors and supporters of FDA08, namely Cankaya University represented by the President of the Board of Trustees Sitki Alp and Rector Professor Ziya B Güvenc, The Scientfic and Technological Research Council of Turkey (TUBITAK) and the IFAC for providing the resources needed to hold the workshop, the invited speakers for sharing their expertise and knowledge of fractional calculus, and the participants for their enthusiastic contributions to the discussions and debates.

Bridging the Vector Calculus Gap

NASA Astrophysics Data System (ADS)

Dray, Tevian; Manogue, Corinne

2003-05-01

As with Britain and America, mathematicians and physicists are separated from each other by a common language. In a nutshell, mathematics is about functions, but physics is about things. For the last several years, we have led an NSF-supported effort to "bridge the vector calculus gap" between mathematics and physics. The unifying theme we have discovered is to emphasize geometric reasoning, not (just) algebraic computation. In this talk, we will illustrate the language differences between mathematicians and physicists, and how we are trying reconcile them in the classroom. For further information about the project go to: http://www.physics.orst.edu/bridge

Integration of CAI into a Freshmen Liberal Arts Math Course in the Community College.

ERIC Educational Resources Information Center

McCall, Michael B.; Holton, Jean L.

1982-01-01

Discusses four computer-assisted-instruction programs used in a college-level mathematics course to introduce computer literacy and improve mathematical skills. The BASIC programs include polynomial functions, trigonometric functions, matrix algebra, and differential calculus. Each program discusses mathematics theory and introduces programming…

University Students' Reading of Their First-Year Mathematics Textbooks

ERIC Educational Resources Information Center

Shepherd, Mary D.; Selden, Annie; Selden, John

2012-01-01

This article reports the observed behaviors and difficulties that 11 precalculus and calculus students exhibited in reading new passages from their mathematics textbooks. To gauge the "effectiveness" of these students' reading, we asked them to attempt straightforward mathematical tasks, based directly on what they had just read. The…

Gestures and Insight in Advanced Mathematical Thinking

ERIC Educational Resources Information Center

Yoon, Caroline; Thomas, Michael O. J.; Dreyfus, Tommy

2011-01-01

What role do gestures play in advanced mathematical thinking? We argue that the role of gestures goes beyond merely communicating thought and supporting understanding--in some cases, gestures can help generate new mathematical insights. Gestures feature prominently in a case study of two participants working on a sequence of calculus activities.…

Factors Shaping Mathematics Lecturers' Service Teaching in Different Departments

ERIC Educational Resources Information Center

Bingolbali, E.; Ozmantar, M. F.

2009-01-01

In this article we focus on university lecturers' approaches to the service teaching and factors that influence their approaches. We present data obtained from the interviews with 19 mathematics and three physics lecturers along with the observations of two mathematics lecturers' calculus courses. The findings show that lecturers' approaches to…

Science modelling in pre-calculus: how to make mathematics problems contextually meaningful

NASA Astrophysics Data System (ADS)

Sokolowski, Andrzej; Yalvac, Bugrahan; Loving, Cathleen

2011-04-01

'Use of mathematical representations to model and interpret physical phenomena and solve problems is one of the major teaching objectives in high school math curriculum' (National Council of Teachers of Mathematics (NCTM), Principles and Standards for School Mathematics, NCTM, Reston, VA, 2000). Commonly used pre-calculus textbooks provide a wide range of application problems. However, these problems focus students' attention on evaluating or solving pre-arranged formulas for given values. The role of scientific content is reduced to provide a background for these problems instead of being sources of data gathering for inducing mathematical tools. Students are neither required to construct mathematical models based on the contexts nor are they asked to validate or discuss the limitations of applied formulas. Using these contexts, the instructor may think that he/she is teaching problem solving, where in reality he/she is teaching algorithms of the mathematical operations (G. Kulm (ed.), New directions for mathematics assessment, in Assessing Higher Order Thinking in Mathematics, Erlbaum, Hillsdale, NJ, 1994, pp. 221-240). Without a thorough representation of the physical phenomena and the mathematical modelling processes undertaken, problem solving unintentionally appears as simple algorithmic operations. In this article, we deconstruct the representations of mathematics problems from selected pre-calculus textbooks and explicate their limitations. We argue that the structure and content of those problems limits students' coherent understanding of mathematical modelling, and this could result in weak student problem-solving skills. Simultaneously, we explore the ways to enhance representations of those mathematical problems, which we have characterized as lacking a meaningful physical context and limiting coherent student understanding. In light of our discussion, we recommend an alternative to strengthen the process of teaching mathematical modelling - utilization of computer-based science simulations. Although there are several exceptional computer-based science simulations designed for mathematics classes (see, e.g. Kinetic Book (http://www.kineticbooks.com/) or Gizmos (http://www.explorelearning.com/)), we concentrate mainly on the PhET Interactive Simulations developed at the University of Colorado at Boulder (http://phet.colorado.edu/) in generating our argument that computer simulations more accurately represent the contextual characteristics of scientific phenomena than their textual descriptions.

The Rhetoric in Mathematics: Newton, Leibniz, the Calculus, and the Rhetorical Force of the Infinitesimal

ERIC Educational Resources Information Center

Reyes, G. Mitchell

2004-01-01

This essay investigates the rhetoric surrounding the appearance of the concept of the infinitesimal in the seventeenth-century Calculus of Sir Isaac Newton and Gottfried Wilhelm Leibniz. Although historians often have positioned rhetoric as a supplemental discipline, this essay shows that rhetoric is the "material" out of which a new and powerful…

A Cognitive Model of College Mathematics Placement

DTIC Science & Technology

1989-08-01

study focused on the precalculus -- calculus placement decision. The Cognitive model uses novel, or analysis level, placement test items in an attempt to...relative to the requirements of a precalculus course. Placement test scores may be partitioned to give analysis and non-analysis subtest scores which can...67 5.1.1 1989 Intercorrelations ....................................................................... 67 5.1.2 1989 Precalculus -Calculus

Cause-Effect Analysis: Improvement of a First Year Engineering Students' Calculus Teaching Model

ERIC Educational Resources Information Center

van der Hoff, Quay; Harding, Ansie

2017-01-01

This study focuses on the mathematics department at a South African university and in particular on teaching of calculus to first year engineering students. The paper reports on a cause-effect analysis, often used for business improvement. The cause-effect analysis indicates that there are many factors that impact on secondary school teaching of…

An Analysis of Diagram Modification and Construction in Students' Solutions to Applied Calculus Problems

ERIC Educational Resources Information Center

Bremigan, Elizabeth George

2005-01-01

In the study reported here, I examined the diagrams that mathematically capable high school students produced in solving applied calculus problems in which a diagram was provided in the problem statement. Analyses of the diagrams contained in written solutions to selected free-response problems from the 1996 BC level Advanced Placement Calculus…

The Effect of Graphing Calculators on Student Achievement in College Algebra and Pre-Calculus Mathematics Courses

ERIC Educational Resources Information Center

Hatem, Neil

2010-01-01

This study investigates the relationship between the use of graphing calculators employed as Type II technology and student achievement, as determined by assessing students' problem solving skills associated with the concept of function, at the college algebra and pre-calculus level. In addition, this study explores the integration of graphing…

The Use of Mathematical Modeling in an Interdisciplinary Setting.

ERIC Educational Resources Information Center

Harshbarger, Ronald J.; Upshaw, Jane

The University of South Carolina at Hilton Head offers a major in Hotel, Restaurant, and Tourism Administration (HRTA). Many students in the HRTA major take both the HRTA Tourism course and a business calculus course. An interdisciplinary research project was designed in an effort to interest these students in the power of calculus as a…

AN ADVANCED PLACEMENT COURSE IN ANALYTIC GEOMETRY AND CALCULUS (MATHEMATICS XV X AP).

ERIC Educational Resources Information Center

DEROLF, JOHN J.; MIENTKA, WALTER E.

THIS TEXT ON ANALYTIC GEOMETRY AND CALCULUS IS A CORRESPONDENCE COURSE DESIGNED FOR ADVANCED PLACEMENT OF HIGH SCHOOL STUDENTS IN COLLEGE. EACH OF THE 21 LESSONS INCLUDES READING ASSIGNMENTS AND LISTS OF PROBLEMS TO BE WORKED. IN ADDITION, SUPPLEMENTARY EXPLANATIONS AND COMMENTS ARE INCLUDED THAT (1) PROVIDE ILLUSTRATIVE EXAMPLES OF CONCEPTS AND…

Cognitive Computer Tools in the Teaching and Learning of Undergraduate Calculus

ERIC Educational Resources Information Center

Borchelt, Nathan

2007-01-01

The purpose of this study was to explore the use of a cognitive computer tool by undergraduate calculus students as they worked cooperatively on mathematical tasks. Specific attention was given to levels of cognitive demand in which the students were engaged as they completed in-class labs with the assistance of MathCAD. Participants were assigned…

Urban Latina/o Undergraduate Students' Negotiations of Identities and Participation in an Emerging Scholars Calculus I Workshop

ERIC Educational Resources Information Center

Oppland-Cordell, Sarah B.

2014-01-01

In this article, the author presents a qualitative multiple case study that explored how two urban Latina/o undergraduate students' emerging mathematical and racial identity constructions influenced their participation in a culturally diverse, Emerging Scholars Program, Calculus I workshop at a predominately White urban university. Drawing on…

The motion behind the symbols: a vital role for dynamism in the conceptualization of limits and continuity in expert mathematics.

PubMed

Marghetis, Tyler; Núñez, Rafael

2013-04-01

The canonical history of mathematics suggests that the late 19th-century "arithmetization" of calculus marked a shift away from spatial-dynamic intuitions, grounding concepts in static, rigorous definitions. Instead, we argue that mathematicians, both historically and currently, rely on dynamic conceptualizations of mathematical concepts like continuity, limits, and functions. In this article, we present two studies of the role of dynamic conceptual systems in expert proof. The first is an analysis of co-speech gesture produced by mathematics graduate students while proving a theorem, which reveals a reliance on dynamic conceptual resources. The second is a cognitive-historical case study of an incident in 19th-century mathematics that suggests a functional role for such dynamism in the reasoning of the renowned mathematician Augustin Cauchy. Taken together, these two studies indicate that essential concepts in calculus that have been defined entirely in abstract, static terms are nevertheless conceptualized dynamically, in both contemporary and historical practice. Copyright © 2013 Cognitive Science Society, Inc.

Using Short Video Lectures to Enhance Mathematics Learning--Experiences on Differential and Integral Calculus Course for Engineering Students

ERIC Educational Resources Information Center

Kinnari-Korpela, Hanna

2015-01-01

Mathematics' skills and knowledge lay the basis for engineering studies. However, the resources targeted to mathematics' teaching are in many cases very limited. During the past years in our university the reduction of mathematics' contact hours has been significant while at the same time the study groups have grown. However, the mathematical…

The State of Proof in Finnish and Swedish Mathematics Textbooks--Capturing Differences in Approaches to Upper-Secondary Integral Calculus

ERIC Educational Resources Information Center

Bergwall, Andreas; Hemmi, Kirsti

2017-01-01

Students' difficulties with proof, scholars' calls for proof to be a consistent part of K-12 mathematics, and the extensive use of textbooks in mathematics classrooms motivate investigations on how proof-related items are addressed in mathematics textbooks. We contribute to textbook research by focusing on opportunities to learn proof-related…

Mathematical biology modules based on modern molecular biology and modern discrete mathematics.

PubMed

Robeva, Raina; Davies, Robin; Hodge, Terrell; Enyedi, Alexander

2010-01-01

We describe an ongoing collaborative curriculum materials development project between Sweet Briar College and Western Michigan University, with support from the National Science Foundation. We present a collection of modules under development that can be used in existing mathematics and biology courses, and we address a critical national need to introduce students to mathematical methods beyond the interface of biology with calculus. Based on ongoing research, and designed to use the project-based-learning approach, the modules highlight applications of modern discrete mathematics and algebraic statistics to pressing problems in molecular biology. For the majority of projects, calculus is not a required prerequisite and, due to the modest amount of mathematical background needed for some of the modules, the materials can be used for an early introduction to mathematical modeling. At the same time, most modules are connected with topics in linear and abstract algebra, algebraic geometry, and probability, and they can be used as meaningful applied introductions into the relevant advanced-level mathematics courses. Open-source software is used to facilitate the relevant computations. As a detailed example, we outline a module that focuses on Boolean models of the lac operon network.

Mathematical Biology Modules Based on Modern Molecular Biology and Modern Discrete Mathematics

PubMed Central

Davies, Robin; Hodge, Terrell; Enyedi, Alexander

2010-01-01

We describe an ongoing collaborative curriculum materials development project between Sweet Briar College and Western Michigan University, with support from the National Science Foundation. We present a collection of modules under development that can be used in existing mathematics and biology courses, and we address a critical national need to introduce students to mathematical methods beyond the interface of biology with calculus. Based on ongoing research, and designed to use the project-based-learning approach, the modules highlight applications of modern discrete mathematics and algebraic statistics to pressing problems in molecular biology. For the majority of projects, calculus is not a required prerequisite and, due to the modest amount of mathematical background needed for some of the modules, the materials can be used for an early introduction to mathematical modeling. At the same time, most modules are connected with topics in linear and abstract algebra, algebraic geometry, and probability, and they can be used as meaningful applied introductions into the relevant advanced-level mathematics courses. Open-source software is used to facilitate the relevant computations. As a detailed example, we outline a module that focuses on Boolean models of the lac operon network. PMID:20810955

Students' Use of Mathematical Representations in Problem Solving.

ERIC Educational Resources Information Center

Santos-Trigo, Manuel

2002-01-01

Documents the experiences of 25 first-year university students with regard to the kinds of tasks calculus instructors should design in order to engage students in mathematical practices that often require the use of a graphing calculator. (MM)

Let's Keep the College in Our Community Colleges: Mathematics for College Transfer.

ERIC Educational Resources Information Center

Curnutt, Larry

Preparing students for transfer to four-year colleges remains a significant part of the mission of most community college mathematicians. For some 30 years, calculus has been synonymous with entry-level college mathematics. Recent educational and technological changes, however, demand that the definition of college-level work in mathematics be…

Studying Teachers' Mathematical Argumentation in the Context of Refuting Students' Invalid Claims

ERIC Educational Resources Information Center

Giannakoulias, Eusthathios; Mastorides, Eleutherios; Potari, Despina; Zachariades, Theodossios

2010-01-01

This study investigates teachers' argumentation aiming to convince students about the invalidity of their mathematical claims in the context of calculus. 18 secondary school mathematics teachers were given three hypothetical scenarios of a student's proof that included an invalid algebraic claim. The teachers were asked to identify possible…

A Conversation with Uri Treisman

ERIC Educational Resources Information Center

Treisman, Uri

2012-01-01

Dr. Uri Treisman, professor of mathematics and public affairs at The University of Texas at Austin and the director of the Charles A. Dana Center, has deep and active roots in mathematics and mathematics education. Dr. Treisman is well known for his early work at the University of California at Berkeley, where he developed the Calculus Workshop…

Using Literacy Strategies to Teach Precalculus and Calculus

ERIC Educational Resources Information Center

Roepke, Tena L.; Gallagher, Debra K.

2015-01-01

Mathematics preservice teachers often complain vehemently about their required content-area reading courses. They ask such questions as, "Why do I have to worry about literacy when I'm going to be a mathematics teacher?" or "How will this ever help me in my mathematics classes?" or "When will I ever have time for…

How High Is the Tramping Track? Mathematising and Applying in a Calculus Model-Eliciting Activity

ERIC Educational Resources Information Center

Yoon, Caroline; Dreyfus, Tommy; Thomas, Michael O. J.

2010-01-01

Two complementary processes involved in mathematical modelling are mathematising a realistic situation and applying a mathematical technique to a given realistic situation. We present and analyse work from two undergraduate students and two secondary school teachers who engaged in both processes during a mathematical modelling task that required…

A Mathematics Support Programme for First-Year Engineering Students

ERIC Educational Resources Information Center

Hillock, Poh Wah; Jennings, Michael; Roberts, Anthony; Scharaschkin, Victor

2013-01-01

This article describes a mathematics support programme at the University of Queensland, targeted at first-year engineering students identified as having a high risk of failing a first-year mathematics course in calculus and linear algebra. It describes how students were identified for the programme and the main features of the programme. The…

A Mixed Methods Analysis of Students' Understanding of Slope and Derivative Concepts and Students' Mathematical Dispositions

ERIC Educational Resources Information Center

Patel, Rita Manubhai

2013-01-01

This dissertation examined understanding of slope and derivative concepts and mathematical dispositions of first-semester college calculus students, who are recent high school graduates, transitioning to university mathematics. The present investigation extends existing research in the following ways. First, based on this investigation, the…

The Mathematical Courses of Pedro Padilla and Etienne Bezout: Teaching Calculus in Eighteenth-Century Spain and France

ERIC Educational Resources Information Center

Blanco, Monica

2013-01-01

The aim of this paper is to provide a cross-national comparative analysis of the introduction of calculus in Spanish and French military educational institutions through the works of Pedro Padilla y Arcos (1724-1807?) and Etienne Bezout (1730-1783), respectively. Both authors developed their educational work in the context of military schools and…

Optimal Strategy in the "Price Is Right" Showcase Showdown: A Module for Students of Calculus and Probability

ERIC Educational Resources Information Center

Swenson, Daniel

2015-01-01

We walk through a module intended for undergraduates in mathematics, with the focus of finding the best strategies for competing in the Showcase Showdown on the game show "The Price Is Right." Students should have completed one semester of calculus, as well as some probability. We also give numerous suggestions for further questions that…

A Mixed-Methods Explanatory Study of the Failure Rate for Freshman STEM Calculus Students

ERIC Educational Resources Information Center

Worthley, Mary R.; Gloeckner, Gene W.; Kennedy, Paul A.

2016-01-01

In this study we aimed to understand who was struggling in freshman calculus courses, and why. Concentrating on the Fall sections of the class, the best predictors for success (R[superscript 2] = 0.4) were placement test results, the student's own appraisal of the quality of mathematics teaching they received in high school, and the Motivated…

Teaching the Concept of Limit by Using Conceptual Conflict Strategy and Desmos Graphing Calculator

ERIC Educational Resources Information Center

Liang, Senfeng

2016-01-01

Although the mathematics community has long accepted the concept of limit as the foundation of modern Calculus, the concept of limit itself has been marginalized in undergraduate Calculus education. In this paper, I analyze the strategy of conceptual conflict to teach the concept of limit with the aid of an online tool--Desmos graphing calculator.…

The Effectiveness of the National Board Certification as It Relates to the Advanced Placement Calculus AB Exam

ERIC Educational Resources Information Center

Antunez, Fernando

2015-01-01

This study compared data related to National Board Certification (NBC) of mathematics teachers in a South Florida school district. Data included 1,162 student scores on the 2014 AP Calculus AB exam, student gender, student grade level, and eligibility for free or reduced price lunch (FRL) status. Teachers completed the Standards' Beliefs…

Exponential Models of Legislative Turnover. [and] The Dynamics of Political Mobilization, I: A Model of the Mobilization Process, II: Deductive Consequences and Empirical Application of the Model. Applications of Calculus to American Politics. [and] Public Support for Presidents. Applications of Algebra to American Politics. Modules and Monographs in Undergraduate Mathematics and Its Applications Project. UMAP Units 296-300.

ERIC Educational Resources Information Center

Casstevens, Thomas W.; And Others

This document consists of five units which all view applications of mathematics to American politics. The first three view calculus applications, the last two deal with applications of algebra. The first module is geared to teach a student how to: 1) compute estimates of the value of the parameters in negative exponential models; and draw…

CIMAC: A Coordinated Introduction to Calculus and Mechanics

NASA Astrophysics Data System (ADS)

Fathe, Laurie; Quinn, Jennifer; McDonald, Michael A.

1997-04-01

CIMAC, new course incorporating Mechanics, Precalculus, and Calculus, targets the growing number of motivated but underprepared students who wish to pursue a major in science or mathematics. Team-taught by a Physicist and a Mathematician, CIMAC, a new course incorporating Mechanics, Precalculus, and Calculus, targets the growing number of motivated but underprepared students who wish to pursue a major in science or mathematics. Team-taught by a Physicist and a Mathematician, the class contains specific content while exploiting the substantial commonality of these subjects. CIMAC also addresses variety of non-content areas, including supplementing basic mathematics and communication skills, accommodating various learning styles, and building student confidence. Specific approaches include class formats; gateway exams; group assignments; emphasis on writing and reading; use of computers and graphing calculators for comprehension, data acquisition, analysis, and modeling; student-led help sessions; and use of the Web http://www.oxy.edu/ departments/math/cimac/ This talk highlights the development of the course and teaching insights and innovations which have arisen from it, and addresses benefits and difficulties of coordinating material and team teaching across disciplinary lines. Finally, it presents data on student success and retention.

Formally verifying Ada programs which use real number types

NASA Technical Reports Server (NTRS)

Sutherland, David

1986-01-01

Formal verification is applied to programs which use real number arithmetic operations (mathematical programs). Formal verification of a program P consists of creating a mathematical model of F, stating the desired properties of P in a formal logical language, and proving that the mathematical model has the desired properties using a formal proof calculus. The development and verification of the mathematical model are discussed.

Improving student learning in calculus through applications

NASA Astrophysics Data System (ADS)

Young, C. Y.; Georgiopoulos, M.; Hagen, S. C.; Geiger, C. L.; Dagley-Falls, M. A.; Islas, A. L.; Ramsey, P. J.; Lancey, P. M.; Straney, R. A.; Forde, D. S.; Bradbury, E. E.

2011-07-01

Nationally only 40% of the incoming freshmen Science, Technology, Engineering and Mathematics (STEM) majors are successful in earning a STEM degree. The University of Central Florida (UCF) EXCEL programme is a National Science Foundation funded STEM Talent Expansion Programme whose goal is to increase the number of UCF STEM graduates. One of the key requirements for STEM majors is a strong foundation in Calculus. To improve student learning in calculus, the EXCEL programme developed two special courses at the freshman level called Applications of Calculus I (Apps I) and Applications of Calculus II (Apps II). Apps I and II are one-credit classes that are co-requisites for Calculus I and II. These classes are teams taught by science and engineering professors whose goal is to demonstrate to students where the calculus topics they are learning appear in upper level science and engineering classes as well as how faculty use calculus in their STEM research programmes. This article outlines the process used in producing the educational materials for the Apps I and II courses, and it also discusses the assessment results pertaining to this specific EXCEL activity. Pre- and post-tests conducted with experimental and control groups indicate significant improvement in student learning in Calculus II as a direct result of the application courses.

Possible Reasons for Students' Ineffective Reading of Their First-Year University Mathematics Textbooks. Technical Report. No. 2011-2

ERIC Educational Resources Information Center

Shepherd, Mary D.; Selden, Annie; Selden, John

2011-01-01

This paper reports the observed behaviors and difficulties that eleven precalculus and calculus students exhibited in reading new passages from their mathematics textbooks. To gauge the effectiveness of these students' reading, we asked them to attempt straightforward mathematical tasks, based directly on what they had just read. These …

Development of an Interactive Mathematics Learning System Based on a Two-Tier Test Diagnostic and Guiding Strategy

ERIC Educational Resources Information Center

Yang, Tzu-Chi; Fu, Hseng-Tz; Hwang, Gwo-Jen; Yang, Stephen J. H.

2017-01-01

Mathematical skills have been recognised as a core competence for engineering and science students. However, learning mathematics has been recognised as a difficult and challenging task for most students, in particular, calculus for first-year students in university. Consequently, the development of effective learning strategies and environments…

Engineering Students' Conceptions of the Derivative and Some Implications for Their Mathematical Education

ERIC Educational Resources Information Center

Bingolbali, E.; Monaghan, J.; Roper, T.

2007-01-01

This paper explores Mechanical Engineering students' conceptions of and preferences for conceptions of the derivative, and their views on mathematics. Data comes from pre-, post- and delayed post-tests, a preference test, interviews with students and an analysis of calculus courses. Data from Mathematics students is used to make comparisons with…

High School Learners' Mental Construction during Solving Optimisation Problems in Calculus: A South African Case Study

ERIC Educational Resources Information Center

Brijlall, Deonarain; Ndlovu, Zanele

2013-01-01

This qualitative case study in a rural school in Umgungundlovu District in KwaZulu-Natal, South Africa, explored Grade 12 learners' mental constructions of mathematical knowledge during engagement with optimisation problems. Ten Grade 12 learners who do pure Mathemat-ics participated, and data were collected through structured activity sheets and…

A Study of Placement and Grade Prediction in First College Mathematics Courses

ERIC Educational Resources Information Center

Madison, Bernard L.; Linde, Cassandra S.; Decker, Blake R.; Rigsby, E. Myron; Dingman, Shannon W.; Stegman, Charles E.

2015-01-01

A college mathematics placement test with 25 basic algebra items and 15 calculus readiness items was administered to 1572 high school seniors, and first college mathematics course grades were obtained for 319 of these students. Test results indicated that more than two thirds of the high school graduates were not college ready, and the test…

Do Screencasts Help to Revise Prerequisite Mathematics? An Investigation of Student Performance and Perception

ERIC Educational Resources Information Center

Loch, Birgit; Jordan, Camilla R.; Lowe, Tim W.; Mestel, Ben D.

2014-01-01

Basic calculus skills that are prerequisites for advanced mathematical studies continue to be a problem for a significant proportion of higher education students. While there are many types of revision material that could be offered to students, in this paper we investigate whether short, narrated video recordings of mathematical explanations…

Space Mathematics, A Resource for Teachers Outlining Supplementary Space-Related Problems in Mathematics.

ERIC Educational Resources Information Center

Reynolds, Thomas D.; And Others

This compilation of 138 problems illustrating applications of high school mathematics to various aspects of space science is intended as a resource from which the teacher may select questions to supplement his regular course. None of the problems require a knowledge of calculus or physics, and solutions are presented along with the problem…

Examinations in the Final Year of Transition to Mathematical Methods Computer Algebra System (CAS)

ERIC Educational Resources Information Center

Leigh-Lancaster, David; Les, Magdalena; Evans, Michael

2010-01-01

2009 was the final year of parallel implementation for Mathematical Methods Units 3 and 4 and Mathematical Methods (CAS) Units 3 and 4. From 2006-2009 there was a common technology-free short answer examination that covered the same function, algebra, calculus and probability content for both studies with corresponding expectations for key…

Connecting Functions in Geometry and Algebra

ERIC Educational Resources Information Center

Steketee, Scott; Scher, Daniel

2016-01-01

One goal of a mathematics education is that students make significant connections among different branches of mathematics. Connections--such as those between arithmetic and algebra, between two-dimensional and three-dimensional geometry, between compass-and-straight-edge constructions and transformations, and between calculus and analytic…

An operator calculus for surface and volume modeling

NASA Technical Reports Server (NTRS)

Gordon, W. J.

1984-01-01

The mathematical techniques which form the foundation for most of the surface and volume modeling techniques used in practice are briefly described. An outline of what may be termed an operator calculus for the approximation and interpolation of functions of more than one independent variable is presented. By considering the linear operators associated with bivariate and multivariate interpolation/approximation schemes, it is shown how they can be compounded by operator multiplication and Boolean addition to obtain a distributive lattice of approximation operators. It is then demonstrated via specific examples how this operator calculus leads to practical techniques for sculptured surface and volume modeling.

Mathematical misconception in calculus 1: Identification and gender difference

NASA Astrophysics Data System (ADS)

Nassir, Asyura Abd; Abdullah, Nur Hidayah Masni; Ahmad, Salimah; Tarmuji, Nor Habibah; Idris, Aminatul Solehah

2017-08-01

A few years of experience of teaching mathematics make us notice that the same types of mistakes are done repeatedly by students. This paper presents an insight into categories of mistakes, how male and female students differ in terms of mistakes that are commonly done and the ability of the students to identify the mistakes. Sample of mistakes were taken from Calculus 1 final exam answer scripts, then it was listed and analyzed. Data analysis revealed that students' misconceptions fall into four categories. The first category is misunderstanding the meaning of brackets, followed by misconception of basic mathematics rules, misconception in notation and misconception in properties of trigonometry. A mistake identification test which consists of ten false mathematical statements was designed based on the mistake done by the previous batch of students that covered topics algebra, trigonometry, index, limit, differentiation and integration. Then, the test was given to students who enrolled in Calculus I course. Respondents of this study were randomly selected among two hundreds engineering students. Data obtained were analyzed using basic descriptive analysis and Chi Square test to capture gender differences in the mistake done for each category. Findings indicate that thirty five percent of the students have the ability to identify the mistakes and make a proper correction for at most two statements. Thirty one percent of the students are able to identify the mistakes but unable to make proper correction. Twenty five percent of the students failed to identify the mistakes in six out of ten false statements. Female students' misconception is more likely in basic mathematics rules compared to male. The findings of this study could serve as baseline information to be stressed in improving teaching and learning mathematics.

The conceptual basis of mathematics in cardiology: (II). Calculus and differential equations.

PubMed

Bates, Jason H T; Sobel, Burton E

2003-04-01

This is the second in a series of four articles developed for the readers of Coronary Artery Disease. Without language ideas cannot be articulated. What may not be so immediately obvious is that they cannot be formulated either. One of the essential languages of cardiology is mathematics. Unfortunately, medical education does not emphasize, and in fact, often neglects empowering physicians to think mathematically. Reference to statistics, conditional probability, multicompartmental modeling, algebra, calculus and transforms is common but often without provision of genuine conceptual understanding. At the University of Vermont College of Medicine, Professor Bates developed a course designed to address these deficiencies. The course covered mathematical principles pertinent to clinical cardiovascular and pulmonary medicine and research. It focused on fundamental concepts to facilitate formulation and grasp of ideas. This series of four articles was developed to make the material available for a wider audience. The articles will be published sequentially in Coronary Artery Disease. Beginning with fundamental axioms and basic algebraic manipulations they address algebra, function and graph theory, real and complex numbers, calculus and differential equations, mathematical modeling, linear system theory and integral transforms and statistical theory. The principles and concepts they address provide the foundation needed for in-depth study of any of these topics. Perhaps of even more importance, they should empower cardiologists and cardiovascular researchers to utilize the language of mathematics in assessing the phenomena of immediate pertinence to diagnosis, pathophysiology and therapeutics. The presentations are interposed with queries (by Coronary Artery Disease abbreviated as CAD) simulating the nature of interactions that occurred during the course itself. Each article concludes with one or more examples illustrating application of the concepts covered to cardiovascular medicine and biology.

Attitudes toward and approaches to learning first-year university mathematics.

PubMed

Alkhateeb, Haitham M; Hammoudi, Lakhdar

2006-08-01

This study examined the relationship for 180 undergraduate students enrolled in a first-year university calculus course between attitudes toward mathematics and approaches to learning mathematics using the Mathematics Attitude Scale and the Approaches to Learning Mathematics Questionnaire, respectively. Regression analyses indicated that scores for the Mathematics Attitude Scale were negatively related to scores for the Surface Approach and accounted for 10.4% of the variance and scores for the Mathematics Attitude Scale were positively related to scores for the Deep Approach to learning mathematics and accounted for 31.7% of the variance.

Student Solution Manual for Essential Mathematical Methods for the Physical Sciences

NASA Astrophysics Data System (ADS)

Riley, K. F.; Hobson, M. P.

2011-02-01

1. Matrices and vector spaces; 2. Vector calculus; 3. Line, surface and volume integrals; 4. Fourier series; 5. Integral transforms; 6. Higher-order ODEs; 7. Series solutions of ODEs; 8. Eigenfunction methods; 9. Special functions; 10. Partial differential equations; 11. Solution methods for PDEs; 12. Calculus of variations; 13. Integral equations; 14. Complex variables; 15. Applications of complex variables; 16. Probability; 17. Statistics.

Essential Mathematical Methods for the Physical Sciences

NASA Astrophysics Data System (ADS)

Riley, K. F.; Hobson, M. P.

2011-02-01

1. Matrices and vector spaces; 2. Vector calculus; 3. Line, surface and volume integrals; 4. Fourier series; 5. Integral transforms; 6. Higher-order ODEs; 7. Series solutions of ODEs; 8. Eigenfunction methods; 9. Special functions; 10. Partial differential equations; 11. Solution methods for PDEs; 12. Calculus of variations; 13. Integral equations; 14. Complex variables; 15. Applications of complex variables; 16. Probability; 17. Statistics; Appendices; Index.

Retention of Differential and Integral Calculus: A Case Study of a University Student in Physical Chemistry

ERIC Educational Resources Information Center

Jukic Matic, Ljerka; Dahl, Bettina

2014-01-01

This paper reports a study on retention of differential and integral calculus concepts of a second-year student of physical chemistry at a Danish university. The focus was on what knowledge the student retained 14 months after the course and on what effect beliefs about mathematics had on the retention. We argue that if a student can quickly…

On the Use of History of Mathematics: An Introduction to Galileo's Study of Free Fall Motion

ERIC Educational Resources Information Center

Ponce Campuzano, Juan Carlos; Matthews, Kelly E.; Adams, Peter

2018-01-01

In this paper, we report on an experimental activity for discussing the concepts of speed, instantaneous speed and acceleration, generally introduced in first year university courses of calculus or physics. Rather than developing the ideas of calculus and using them to explain these basic concepts for the study of motion, we led 82 first year…

Underground Mathematics

ERIC Educational Resources Information Center

Hadlock, Charles R

2013-01-01

The movement of groundwater in underground aquifers is an ideal physical example of many important themes in mathematical modeling, ranging from general principles (like Occam's Razor) to specific techniques (such as geometry, linear equations, and the calculus). This article gives a self-contained introduction to groundwater modeling with…

On Mathematical Proving

NASA Astrophysics Data System (ADS)

Stefaneas, Petros; Vandoulakis, Ioannis M.

2015-12-01

This paper outlines a logical representation of certain aspects of the process of mathematical proving that are important from the point of view of Artificial Intelligence. Our starting-point is the concept of proof-event or proving, introduced by Goguen, instead of the traditional concept of mathematical proof. The reason behind this choice is that in contrast to the traditional static concept of mathematical proof, proof-events are understood as processes, which enables their use in Artificial Intelligence in such contexts, in which problem-solving procedures and strategies are studied. We represent proof-events as problem-centered spatio-temporal processes by means of the language of the calculus of events, which captures adequately certain temporal aspects of proof-events (i.e. that they have history and form sequences of proof-events evolving in time). Further, we suggest a "loose" semantics for the proof-events, by means of Kolmogorov's calculus of problems. Finally, we expose the intented interpretations for our logical model from the fields of automated theorem-proving and Web-based collective proving.

Approach to mathematics in textbooks at tertiary level - exploring authors' views about their texts

NASA Astrophysics Data System (ADS)

Randahl, Mira

2012-10-01

The aim of this article is to present and discuss some results from an inquiry into mathematics textbooks authors' visions about their texts and approaches they choose when new concepts are introduced. Authors' responses are discussed in relation to results about students' difficulties with approaching calculus reported by previous research. A questionnaire has been designed and sent to seven authors of the most used calculus textbooks in Norway and four authors have responded. The responses show that the authors mainly view teaching in terms of transmission so they focus mainly on getting the mathematical content correct and 'clear'. The dominant view is that the textbook is intended to help the students to learn by explaining and clarifying. The authors prefer the approach to introduce new concepts based on the traditional way of perceiving mathematics as a system of definitions, examples and exercises. The results of this study may enhance our understanding of the role of the textbook at tertiary level. They may also form a foundation for further research.

The Mathematical Courses of Pedro Padilla and Étienne Bézout: Teaching Calculus in Eighteenth-Century Spain and France

NASA Astrophysics Data System (ADS)

Blanco, Mónica

2013-04-01

The aim of this paper is to provide a cross-national comparative analysis of the introduction of calculus in Spanish and French military educational institutions through the works of Pedro Padilla y Arcos (1724-1807?) and Étienne Bézout (1730-1783), respectively. Both authors developed their educational work in the context of military schools and academies. Padilla's Curso Militar de Mathematicas (1753-1756) was the first work published in Spain which introduced the teaching of calculus in formal education. Bézout's Cours de Mathématiques (1764-1769) was the first work on calculus explicitly addressed to French military students and can be considered a representative of the canonical knowledge on eighteenth-century mathematics, both in France and abroad. Eighteenth-century Spain has traditionally been regarded as a country in the periphery whose scientific culture and education were pervaded by French science and education. This centre-periphery framework is often represented by a static model of one-way transmission from the centre to the periphery. A crossnational comparative analysis can help revisit this monolithic centre-periphery framework. A recent historiographical stream places the emphasis on appropriation, hence moving away from the idea of passive reception. In my paper I focus on the reading and writing of educational books, as practices which contribute actively to the development and circulation of knowledge. To assist the analysis, I explore the differences in communication practices in each case, in contents and approaches, and in particular, I give special attention to their inspiration in mathematical streams other than the French standpoint.

Mathematical Concepts and Proofs from Nicole Oresme: Using the History of Calculus to Teach Mathematics

ERIC Educational Resources Information Center

Babb, Jeff

2005-01-01

This paper examines the mathematical work of the French bishop, Nicole Oresme (c. 1323-1382), and his contributions towards the development of the concept of graphing functions and approaches to investigating infinite series. The historical importance and pedagogical value of his work will be considered in the context of an undergraduate course on…

Tracking the Success of Pre-College Algebra Workshop Students in Subsequent College Mathematics Classes

ERIC Educational Resources Information Center

Fuller, Edgar; Deshler, Jessica M.; Kuhn, Betsy; Squire, Douglas

2014-01-01

In 2007 the Department of Mathematics at our institution began developing a placement process designed to identify at-risk students entering mathematics courses at the College Algebra and Calculus levels. Major changes in our placement testing process and the resulting interventions for at-risk students were put in place in Fall of 2008. At the…

Mathematical Preparedness for Tertiary Mathematics--A Need for Focused Intervention in the First Year?

ERIC Educational Resources Information Center

Du Preez, Jeanetta; Steyn, Tobia; Owen, Rina

2008-01-01

Ongoing action research at the University of Pretoria investigates first-year students' preparedness for a study in calculus. In 2005 first-year engineering students completed a mathematics diagnostic survey at the beginning and end of the year. In this article the results of the 2005 survey are compared with the students' final school marks in…

Critical Analysis of the Mathematical Formalism of Theoretical Physics. II. Foundations of Vector Calculus

NASA Astrophysics Data System (ADS)

Kalanov, Temur Z.

2014-03-01

A critical analysis of the foundations of standard vector calculus is proposed. The methodological basis of the analysis is the unity of formal logic and of rational dialectics. It is proved that the vector calculus is incorrect theory because: (a) it is not based on a correct methodological basis - the unity of formal logic and of rational dialectics; (b) it does not contain the correct definitions of ``movement,'' ``direction'' and ``vector'' (c) it does not take into consideration the dimensions of physical quantities (i.e., number names, denominate numbers, concrete numbers), characterizing the concept of ''physical vector,'' and, therefore, it has no natural-scientific meaning; (d) operations on ``physical vectors'' and the vector calculus propositions relating to the ''physical vectors'' are contrary to formal logic.

Cutting Cakes Carefully

ERIC Educational Resources Information Center

Hill, Theodore P.; Morrison, Kent E.

2010-01-01

This paper surveys the fascinating mathematics of fair division, and provides a suite of examples using basic ideas from algebra, calculus, and probability which can be used to examine and test new and sometimes complex mathematical theories and claims involving fair division. Conversely, the classical cut-and-choose and moving-knife algorithms…

Introductory Life Science Mathematics and Quantitative Neuroscience Courses

ERIC Educational Resources Information Center

Duffus, Dwight; Olifer, Andrei

2010-01-01

We describe two sets of courses designed to enhance the mathematical, statistical, and computational training of life science undergraduates at Emory College. The first course is an introductory sequence in differential and integral calculus, modeling with differential equations, probability, and inferential statistics. The second is an…

Key Concept Mathematics and Management Science Models

ERIC Educational Resources Information Center

Macbeth, Thomas G.; Dery, George C.

1973-01-01

The presentation of topics in calculus and matrix algebra to second semester freshmen along with a treatment of exponential and power functions would permit them to cope with a significant portion of the mathematical concepts that comprise the essence of several disciplines in a business school curriculum. (Author)

Some Mathematics and Physics of Ball Games.

ERIC Educational Resources Information Center

Hughes, D. E.

1985-01-01

Gives examples on the applications of arithmetic, geometry, and some calculus, vector algebra, and mechanics to ball games. Suggestions for further interesting investigations are provided together with references to other articles and books on applications of mathematics and physics to ball games and sports in general. (JN)

By Doctrines Fashioned to the Varying Hour or the Calculus of Horrors

ERIC Educational Resources Information Center

Srinivasan, V. K.

2002-01-01

A deliberate attempt is made in Business Mathematics oriented text books as well as in some reform calculus oriented text books to interpret the derivative f[prime](a) of a function y = f(x) at the value x = a as the change in the y-value of the function per "unit" of change in the x-value. This note questions the above interpretation and suggests…

Students attitude towards calculus subject: Bumiputera case-study

NASA Astrophysics Data System (ADS)

Awang, Noorehan; Ilias, Mohd Rijal; Che Hussain, Wan Siti Esah; Mokhtar, Siti Fairus

2013-04-01

Mathematics has always become the most dislike subject among other subjects in school. Study showed that attitudes of students in science subjects such as mathematics were closely related to how they solve problems, accessing ideas and making a right decision. According to another study on mathematics achievement of eighth grade students in Malaysia, mathematics grades among bumiputera students was lower when compared to other races such as Chinese and Indians. The poor performance was due to their attitude and pre-conceived ideas towards the subject. Therefore, this study was designed todetermine the criteria and subcriteria that were considered important in measuring students' attitude toward mathematics among the bumiputeras. Factor analysis was carried out to identify the groups among criterion. Instrument used to measure mathematics attitude was Test of Mathematics Related Attitude (TOMRA) which measured student attitudes in four criteria: normality of mathematics, attitudes towards mathematics inquiry, adoption of mathematics attitude and enjoyment of mathematics lessons. The target population of this study was all computer science and quantitative science students who enrolled Calculus subject in UiTM Kedah. Findings shows that there are two criteria that influenced students attitude toward mathematics namely normality of mathematics with eleven subcriteria and enjoyment of mathematics with eight subcriteria. From the analysis it shows that the total percentage of variation explained is 35.071% with 0.837 Cronbach's alpha reliability test. The findings will help the lecturers, parents and society to consider what action should be taken to install interest and positive attitude of bumiputera students towards mathematics and thus improve their achievement.

On flipping first-semester calculus: a case study

NASA Astrophysics Data System (ADS)

Petrillo, Joseph

2016-05-01

High failure rates in calculus have plagued students, teachers, and administrators for decades, while science, technology, engineering, and mathematics programmes continue to suffer from low enrollments and high attrition. In an effort to affect this reality, some educators are 'flipping' (or inverting) their classrooms. By flipping, we mean administering course content outside of the classroom and replacing the traditional in-class lectures with discussion, practice, group work, and other elements of active learning. This paper presents the major results from a three-year study of a flipped, first-semester calculus course at a small, comprehensive, American university with a well-known engineering programme. The data we have collected help quantify the positive and substantial effects of our flipped calculus course on failure rates, scores on the common final exam, student opinion of calculus, teacher impact on measurable outcomes, and success in second-semester calculus. While flipping may not be suitable for every teacher, every student, and in every situation, this report provides some evidence that it may be a viable option for those seeking an alternative to the traditional lecture model.

A useful demonstration of calculus in a physics high school laboratory

NASA Astrophysics Data System (ADS)

Alvarez, Gustavo; Schulte, Jurgen; Stockton, Geoffrey; Wheeler, David

2018-01-01

The real power of calculus is revealed when it is applied to actual physical problems. In this paper, we present a calculus inspired physics experiment suitable for high school and undergraduate programs. A model for the theory of the terminal velocity of a falling body subject to a resistive force is developed and its validity tested in an experiment of a falling magnet in a column of self-induced eddy currents. The presented method combines multiple physics concepts such as 1D kinematics, classical mechanics, electromagnetism and non-trivial mathematics. It offers the opportunity for lateral as well as project-based learning.

Hamilton-Jacobi-Bellman equations and approximate dynamic programming on time scales.

PubMed

Seiffertt, John; Sanyal, Suman; Wunsch, Donald C

2008-08-01

The time scales calculus is a key emerging area of mathematics due to its potential use in a wide variety of multidisciplinary applications. We extend this calculus to approximate dynamic programming (ADP). The core backward induction algorithm of dynamic programming is extended from its traditional discrete case to all isolated time scales. Hamilton-Jacobi-Bellman equations, the solution of which is the fundamental problem in the field of dynamic programming, are motivated and proven on time scales. By drawing together the calculus of time scales and the applied area of stochastic control via ADP, we have connected two major fields of research.

Shaking up Pre-Calculus: Incorporating Engineering into K-12 Curricula

ERIC Educational Resources Information Center

Sabo, Chelsea; Burrows, Andrea; Childers, Lois

2014-01-01

Projects highlighting Science, Technology, Engineering, and Mathematics (STEM) education in high schools have promoted student interest in engineering-related fields and enhanced student understanding of mathematics and science concepts. The Science and Technology Enhancement Program (Project STEP), funded by a NSF GK-12 grant at the University of…

An Activity to Encourage Writing in Mathematics

ERIC Educational Resources Information Center

Van Dyke, Frances; Malloy, Elizabeth J.; Stallings, Virginia

2014-01-01

This article discusses an activity designed to encourage writing to learn in mathematics. There were three stages of data collection. An assessment, requiring basic algebra only, was completed by 118 undergraduates from statistics and calculus courses. Students were given summaries of all participant responses, along with the correct answers.…

Chaos: A Mathematical Introduction

NASA Astrophysics Data System (ADS)

Banks, John; Dragan, Valentina; Jones, Arthur

2003-06-01

This text presents concepts on chaos in discrete time dynamics that are accessible to anyone who has taken a first course in undergraduate calculus. Retaining its commitment to mathematical integrity, the book, originating in a popular one-semester middle level undergraduate course, constitutes the first elementary presentation of a traditionally advanced subject.

Establishing an Explanatory Model for Mathematics Identity

ERIC Educational Resources Information Center

Cribbs, Jennifer D.; Hazari, Zahra; Sonnert, Gerhard; Sadler, Philip M.

2015-01-01

This article empirically tests a previously developed theoretical framework for mathematics identity based on students' beliefs. The study employs data from more than 9,000 college calculus students across the United States to build a robust structural equation model. While it is generally thought that students' beliefs about their own competence…

A Flexible, Extensible Online Testing System for Mathematics

ERIC Educational Resources Information Center

Passmore, Tim; Brookshaw, Leigh; Butler, Harry

2011-01-01

An online testing system developed for entry-skills testing of first-year university students in algebra and calculus is described. The system combines the open-source computer algebra system "Maxima" with computer scripts to parse student answers, which are entered using standard mathematical notation and conventions. The answers can…

Mathematical Building-Blocks in Engineering Mechanics

ERIC Educational Resources Information Center

Boyajian, David M.

2007-01-01

A gamut of mathematical subjects and concepts are taught within a handful of courses formally required of the typical engineering student who so often questions the relevancy of being bound to certain lower-division prerequisites. Basic classes at the undergraduate level, in this context, include: Integral and Differential Calculus, Differential…

Mathematical Modelling in Engineering: An Alternative Way to Teach Linear Algebra

ERIC Educational Resources Information Center

Domínguez-García, S.; García-Planas, M. I.; Taberna, J.

2016-01-01

Technological advances require that basic science courses for engineering, including Linear Algebra, emphasize the development of mathematical strengths associated with modelling and interpretation of results, which are not limited only to calculus abilities. Based on this consideration, we have proposed a project-based learning, giving a dynamic…

Maximum Pre-Angiogenic Tumor Size

ERIC Educational Resources Information Center

Erickson, Amy H. Lin

2010-01-01

This material has been used twice as an out-of-class project in a mathematical modeling class, the first elective course for mathematics majors. The only prerequisites for this course were differential and integral calculus, but all students had been exposed to differential equations, and the project was assigned during discussions about solving…

Assessing Mathematics Automatically Using Computer Algebra and the Internet

ERIC Educational Resources Information Center

Sangwin, Chris

2004-01-01

This paper reports some recent developments in mathematical computer-aided assessment which employs computer algebra to evaluate students' work using the Internet. Technical and educational issues raised by this use of computer algebra are addressed. Working examples from core calculus and algebra which have been used with first year university…

The Mathematics Textbook at Tertiary Level as Curriculum Material--Exploring the Teacher's Decision-Making Process

ERIC Educational Resources Information Center

Randahl, Mira

2016-01-01

This paper reports on a study about how the mathematics textbook was perceived and used by the teacher in the context of a calculus part of a basic mathematics course for first-year engineering students. The focus was on the teacher's choices and the use of definitions, examples and exercises in a sequence of lectures introducing the derivative…

Space Mathematics: A Resource for Secondary School Teachers

NASA Technical Reports Server (NTRS)

Kastner, Bernice

1985-01-01

A collection of mathematical problems related to NASA space science projects is presented. In developing the examples and problems, attention was given to preserving the authenticity and significance of the original setting while keeping the level of mathematics within the secondary school curriculum. Computation and measurement, algebra, geometry, probability and statistics, exponential and logarithmic functions, trigonometry, matrix algebra, conic sections, and calculus are among the areas addressed.

Profile of Metacognition of Mathematics Pre-Service Teachers in Understanding the Concept of Integral Calculus with Regard Gender Differences

NASA Astrophysics Data System (ADS)

Misu, L.; Budayasa, I. K.; Lukito, A.

2018-01-01

This research is to describe metacognition profile of female and male mathematics’ pre-service teachers in understanding the concept of integral calculus. The subjects of this study are one female and 1 male mathematics’ pre-service teachers who have studied integral calculus. This research type is an explorative study with the qualitative approach. The main data collection of this research was obtained by using Interview technique. In addition, there are supporting data which is the result of the written work of research subjects (SP) in understanding the question of integral calculus. The results of this study are as follows: There is a difference in metacognition profiles between male and female mathematics’ pre-service teachers in the understanding concept of integral calculus in the interpreting category, especially the definite integral concept. While in the category of exemplifying, there is no difference in metacognition profile between male and female mathematics’ pre-service teachers either the definite integral concept and the indefinite integral concept.

Case Study: Students’ Symbolic Manipulation in Calculus Among UTHM Students

NASA Astrophysics Data System (ADS)

Ali, Maselan; Sufahani, Suliadi; Ahmad, Wan N. A. W.; Ghazali Kamardan, M.; Saifullah Rusiman, Mohd; Che-Him, Norziha

2018-04-01

Words are symbols representing certain aspects of mathematics. The main purpose of this study is to gain insight into students’ symbolic manipulation in calculus among UTHM students. This study make use the various methods in collecting data which are documentation, pilot study, written test and follow up individual interviews. Hence, the results analyzed and interpreted based on action-process-object-schema framework which is based on Piaget’s ideas of reflective abstraction, the concept of relational and instrumental understanding and the zone of proximal development idea. The students’ reply in the interview session is analyzed and then the overall performance is discussed briefly to relate with the students flexibility in symbolic manipulation in linking to the graphical idea, the students interpretation towards different symbolic structure in calculus and the problem that related to overgeneralization in their calculus problems solving.

Science and Society Test for Scientists: Transportation

ERIC Educational Resources Information Center

Hafemeister, David

1976-01-01

Presents numerous questions concerning transportation systems, energy consumption, noise, air pollution and other transportation oriented topics. Solutions are provided using undergraduate pre-calculus mathematics. (CP)

Improving basic math skills through integrated dynamic representation strategies.

PubMed

González-Castro, Paloma; Cueli, Marisol; Cabeza, Lourdes; Álvarez-García, David; Rodríguez, Celestino

2014-01-01

In this paper, we analyze the effectiveness of the Integrated Dynamic Representation strategy (IDR) to develop basic math skills. The study involved 72 students, aged between 6 and 8 years. We compared the development of informal basic skills (numbers, comparison, informal calculation, and informal concepts) and formal (conventionalisms, number facts, formal calculus, and formal concepts) in an experimental group (n = 35) where we applied the IDR strategy and in a Control group (n = 37) in order to identify the impact of the procedure. The experimental group improved significantly in all variables except for number facts and formal calculus. It can therefore be concluded that IDR favors the development of the skills more closely related to applied mathematics than those related to automatic mathematics and mental arithmetic.

Applied Mathematical Methods in Theoretical Physics

NASA Astrophysics Data System (ADS)

Masujima, Michio

2005-04-01

All there is to know about functional analysis, integral equations and calculus of variations in a single volume. This advanced textbook is divided into two parts: The first on integral equations and the second on the calculus of variations. It begins with a short introduction to functional analysis, including a short review of complex analysis, before continuing a systematic discussion of different types of equations, such as Volterra integral equations, singular integral equations of Cauchy type, integral equations of the Fredholm type, with a special emphasis on Wiener-Hopf integral equations and Wiener-Hopf sum equations. After a few remarks on the historical development, the second part starts with an introduction to the calculus of variations and the relationship between integral equations and applications of the calculus of variations. It further covers applications of the calculus of variations developed in the second half of the 20th century in the fields of quantum mechanics, quantum statistical mechanics and quantum field theory. Throughout the book, the author presents over 150 problems and exercises -- many from such branches of physics as quantum mechanics, quantum statistical mechanics, and quantum field theory -- together with outlines of the solutions in each case. Detailed solutions are given, supplementing the materials discussed in the main text, allowing problems to be solved making direct use of the method illustrated. The original references are given for difficult problems. The result is complete coverage of the mathematical tools and techniques used by physicists and applied mathematicians Intended for senior undergraduates and first-year graduates in science and engineering, this is equally useful as a reference and self-study guide.

Clickers and Classroom Voting in a Transition to Advanced Mathematics Course

ERIC Educational Resources Information Center

Lockard, Shannon R.; Metcalf, Rebecca C.

2015-01-01

Clickers and classroom voting are used across a number of disciplines in a variety of institutions. There are several papers that describe the use of clickers in mathematics classrooms such as precalculus, calculus, statistics, and even differential equations. This paper describes a method of incorporating clickers and classroom voting in a…

Object-Spatial Visualization and Verbal Cognitive Styles, and Their Relation to Cognitive Abilities and Mathematical Performance

ERIC Educational Resources Information Center

Haciomeroglu, Erhan Selcuk

2016-01-01

The present study investigated the object-spatial visualization and verbal cognitive styles among high school students and related differences in spatial ability, verbal-logical reasoning ability, and mathematical performance of those students. Data were collected from 348 students enrolled in Advanced Placement calculus courses at six high…

Students' Understanding of Mathematical Expressions in Physical Chemistry Contexts: An Analysis Using Sherin's Symbolic Forms

ERIC Educational Resources Information Center

Becker, Nicole; Towns, Marcy

2012-01-01

Undergraduate physical chemistry courses require students to be proficient in calculus in order to develop an understanding of thermodynamics concepts. Here we present the findings of a study that examines student understanding of mathematical expressions, including partial derivative expressions, in two undergraduate physical chemistry courses.…

Obstacles to Mathematization in Physics: The Case of the Differential

ERIC Educational Resources Information Center

López-Gay, R.; Martinez Sáez, J.; Martinez Torregrosa, J.

2015-01-01

The process of the mathematization of physical situations through differential calculus requires an understanding of the justification for and the meaning of the differential in the context of physics. In this work, four different conceptions about the differential in physics are identified and assessed according to their utility for the…

Addressing the Standards for Mathematical Practice in a Calculus Class

ERIC Educational Resources Information Center

Pilgrim, Mary E.

2014-01-01

The Common Core State Standards (CCSS) provide teachers with the expectations and requirements that are meant to prepare K-12 students for college and the workforce (CCSSI 2010b). The Common Core State Standards for Mathematical Practice (SMPs) emphasize the development of skills and conceptual understanding for students to become proficient in…

r dr r: Engaging Students with Significant Mathematical Content from The Simpsons

ERIC Educational Resources Information Center

Greenwald, Sarah J.; Nestler, Andrew

2004-01-01

"The Simpsons" is an ideal source of fun ways to introduce important mathematical concepts, motivate students, and reduce math anxiety. We discuss examples from "The Simpsons" related to calculus, geometry, and number theory that we have incorporated into the classroom. We explore student reactions and educational benefits and difficulties…

Calculus-Based Mathematics: An Australian Endangered Species?

ERIC Educational Resources Information Center

Maltas, Dimitrios; Prescott, Anne

2014-01-01

Many people are discussing the issues surrounding mathematics at all levels of education. Politicians, parents, students, universities, education departments all have a view about what the problem is and all have ideas about what should happen. This article represents a synthesis of the issues and implications of one of the problems evident in…

The Complexities of a Lesson Study in a Dutch Situation: Mathematics Teacher Learning

ERIC Educational Resources Information Center

Verhoef, Nellie; Tall, David; Coenders, Fer; van Smaalen, Daan

2014-01-01

This study combines the Japanese lesson study approach and mathematics teachers' professional development. The first year of a 4-year project in which 3 Dutch secondary school teachers worked cooperatively on introducing making sense of the calculus is reported. The analysis focusses on instrumental and relational student understanding of…

In Their Own Words: Getting Pumped for Calculus

ERIC Educational Resources Information Center

Cardetti, Fabiana; McKenna, P. J.

2011-01-01

By understanding what motivates students to learn mathematics, instructors can adapt their teaching strategies to help students achieve a level of engagement conducive to learning mathematics. There is a substantial body of research on motivations and academic achievement at the elementary and high school levels, but less is known at the college…

Three Different Teaching Approaches in Pre-Calculus Bridging Mathematics

ERIC Educational Resources Information Center

Miller-Reilly, Barbara

2007-01-01

During the past decade three different bridging mathematics courses have been offered at the University of Auckland. A case study approach was used to investigate the effectiveness of these courses: two larger courses and one individual study programme. A different teaching approach, by committed experienced teachers, was used in each course. The…

Mathematics in Chemistry: Indeterminate Forms and Their Meaning

ERIC Educational Resources Information Center

Segurado, Manuel A. P.; Silva, Margarida F. B.; Castro, Rita

2011-01-01

The mathematical language and its tools are complementary to the formalism in chemistry, in particular at an advanced level. It is thus crucial, for its understanding, that students acquire a solid knowledge in Calculus and that they know how to apply it. The frequent occurrence of indeterminate forms in multiple areas, particularly in Physical…

Secondary Schools Curriculum Guide, Mathematics, Grades 10-12, Levels 87-112.

ERIC Educational Resources Information Center

Rogers, Arnold R., Ed.; And Others

Behavioral objectives for geometry, algebra, computer mathematics, trigonometry, analytic geometry, calculus, and probability are specified for grades 10 through 12. General objectives are stated for major areas under each topic and are followed by a list of specific objectives for that area. This work was prepared under an ESEA Title III…

Enhancing Mathematical Communication for Virtual Math Teams

ERIC Educational Resources Information Center

Stahl, Gerry; Çakir, Murat Perit; Weimar, Stephen; Weusijana, Baba Kofi; Ou, Jimmy Xiantong

2010-01-01

The Math Forum is an online resource center for pre-algebra, algebra, geometry and pre-calculus. Its Virtual Math Teams (VMT) service provides an integrated web-based environment for small teams of people to discuss math and to work collaboratively on math problems or explore interesting mathematical micro-worlds together. The VMT Project studies…

"MathePraxis"--Connecting First-Year Mathematics with Engineering Applications

ERIC Educational Resources Information Center

Harterich, Jorg; Kiss, Christine; Rooch, Aeneas; Monnigmann, Martin; Darup, Moritz Schulze; Span, Roland

2012-01-01

First-year engineering students often complain about their mathematics courses as the significance of the difficult and abstract calculus to their field of study remains unclear. We report on the project "MathePraxis", a feasibility study which was designed as a means to give first-year students some impression about the use of…

A Mathematics Entrance Exam for General (Non-Majors) Physics

ERIC Educational Resources Information Center

Chediak, Alex

2010-01-01

In a previous issue of "The Physics Teacher", John Hubisz explained how a mathematics background check has been used at three different colleges to determine the appropriate physics sequence for incoming students. Based on their performance, students are placed into either calculus-based physics (CBP), algebra-trig physics (ATP), or a year of…

Proceedings: Summer Conference for College Teachers on Applied Mathematics, University of Missouri-Rolla, 1971.

ERIC Educational Resources Information Center

Committee on the Undergraduate Program in Mathematics, Berkeley, CA.

Proceedings from four sessions of the Summer Conference for College Teachers on Applied Mathematics are presented. The four sessions were: (1) Applications of Elementary Calculus, (2) Applications of Linear Algebra, (3) Applications of Elementary Differential Equations, and (4) Applications of Probability and Statistics. Nine lectures were given…

Perceived Utility of Typesetting Homework in Post-Calculus Mathematics Courses

ERIC Educational Resources Information Center

Quinlan, James; Tennenhouse, Craig

2016-01-01

Too often our students submit incomplete homework that is disorganized, unclear, and nonlinear. Typesetting with LATEX, although time consuming for those new to the software, strengthens communication by forcing organization and proper notation required by the precise, formal language of mathematics. In this manuscript we report on a study of 42…

Maple (Computer Algebra System) in Teaching Pre-Calculus: Example of Absolute Value Function

ERIC Educational Resources Information Center

Tuluk, Güler

2014-01-01

Modules in Computer Algebra Systems (CAS) make Mathematics interesting and easy to understand. The present study focused on the implementation of the algebraic, tabular (numerical), and graphical approaches used for the construction of the concept of absolute value function in teaching mathematical content knowledge along with Maple 9. The study…

Mathematical Modeling of Diverse Phenomena

NASA Technical Reports Server (NTRS)

Howard, J. C.

1979-01-01

Tensor calculus is applied to the formulation of mathematical models of diverse phenomena. Aeronautics, fluid dynamics, and cosmology are among the areas of application. The feasibility of combining tensor methods and computer capability to formulate problems is demonstrated. The techniques described are an attempt to simplify the formulation of mathematical models by reducing the modeling process to a series of routine operations, which can be performed either manually or by computer.

Solving multi-customer FPR model with quality assurance and discontinuous deliveries using a two-phase algebraic approach.

PubMed

Chiu, Yuan-Shyi Peter; Chou, Chung-Li; Chang, Huei-Hsin; Chiu, Singa Wang

2016-01-01

A multi-customer finite production rate (FPR) model with quality assurance and discontinuous delivery policy was investigated in a recent paper (Chiu et al. in J Appl Res Technol 12(1):5-13, 2014) using differential calculus approach. This study employs mathematical modeling along with a two-phase algebraic method to resolve such a specific multi-customer FPR model. As a result, the optimal replenishment lot size and number of shipments can be derived without using the differential calculus. Such a straightforward method may assist practitioners who with insufficient knowledge of calculus in learning and managing the real multi-customer FPR systems more effectively.

Students' difficulties with vector calculus in electrodynamics

NASA Astrophysics Data System (ADS)

Bollen, Laurens; van Kampen, Paul; De Cock, Mieke

2015-12-01

Understanding Maxwell's equations in differential form is of great importance when studying the electrodynamic phenomena discussed in advanced electromagnetism courses. It is therefore necessary that students master the use of vector calculus in physical situations. In this light we investigated the difficulties second year students at KU Leuven encounter with the divergence and curl of a vector field in mathematical and physical contexts. We have found that they are quite skilled at doing calculations, but struggle with interpreting graphical representations of vector fields and applying vector calculus to physical situations. We have found strong indications that traditional instruction is not sufficient for our students to fully understand the meaning and power of Maxwell's equations in electrodynamics.

Handbook of applied mathematics for engineers and scientists

DOE Office of Scientific and Technical Information (OSTI.GOV)

Kurtz, M.

1991-12-31

This book is intended to be reference for applications of mathematics in a wide range of topics of interest to engineers and scientists. An unusual feature of this book is that it covers a large number of topics from elementary algebra, trigonometry, and calculus to computer graphics and cybernetics. The level of mathematics covers high school through about the junior level of an engineering curriculum in a major univeristy. Throughout, the emphasis is on applications of mathematics rather than on rigorous proofs.

A structural equation modeling of executive functions, IQ and mathematical skills in primary students: Differential effects on number production, mental calculus and arithmetical problems.

PubMed

Arán Filippetti, Vanessa; Richaud, María Cristina

2017-10-01

Though the relationship between executive functions (EFs) and mathematical skills has been well documented, little is known about how both EFs and IQ differentially support diverse math domains in primary students. Inconsistency of results may be due to the statistical techniques employed, specifically, if the analysis is conducted with observed variables, i.e., regression analysis, or at the latent level, i.e., structural equation modeling (SEM). The current study explores the contribution of both EFs and IQ in mathematics through an SEM approach. A total of 118 8- to 12-year-olds were administered measures of EFs, crystallized (Gc) and fluid (Gf) intelligence, and math abilities (i.e., number production, mental calculus and arithmetical problem-solving). Confirmatory factor analysis (CFA) offered support for the three-factor solution of EFs: (1) working memory (WM), (2) shifting, and (3) inhibition. Regarding the relationship among EFs, IQ and math abilities, the results of the SEM analysis showed that (i) WM and age predict number production and mental calculus, and (ii) shifting and sex predict arithmetical problem-solving. In all of the SEM models, EFs partially or totally mediated the relationship between IQ, age and math achievement. These results suggest that EFs differentially supports math abilities in primary-school children and is a more significant predictor of math achievement than IQ level.

Proceedings of the Annual Conference of the International Group for the Psychology of Mathematics Education with the North American Chapter 12th PME-NA Conference (14th, Mexico, July 15-20, 1990), Volume 1.

ERIC Educational Resources Information Center

Booker, George, Ed.; Cobb, Paul, Ed.; de Mendicuti, Teresa N., Ed.

This proceedings of the annual conference of the International Group for the Psychology of Mathematics Education (PME) includes the following papers: "The Knowledge of Cats: Epistemological Foundations of Mathematics Education" (R.B. Davis) and "PME Algebra Research: A Working Perspective" (E. Filloy); "Some Misconceptions in Calculus: Anecdotes…

Challenges in assessing college students' conception of duality: the case of infinity

NASA Astrophysics Data System (ADS)

Babarinsa-Ochiedike, Grace Olutayo

Interpreting students' views of infinity posits a challenge for researchers due to the dynamic nature of the conception. There is diversity and variation among students' process-object perceptions. The fluctuations between students' views however reveal an undeveloped duality conception. This study examined college students' conception of duality in understanding and representing infinity with the intent to design strategies that could guide researchers in categorizing students' views of infinity into different levels. Data for the study were collected from N=238 college students enrolled in Calculus sequence courses (Pre-Calculus, Calculus I through Calculus III) at one of the southwestern universities in the U.S. using self-report questionnaires and semi-structured individual task-based interviews. Data was triangulated using multiple measures analyzed by three independent experts using self-designed coding sheets to assess students' externalization of the duality conception of infinity. Results of this study reveal that college students' experiences in traditional Calculus sequence courses are not supportive of the development of duality conception. On the contrary, it strengthens the singularity perspective on fundamental ideas of mathematics such as infinity. The study also found that coding and assessing college students' conception of duality is a challenging and complex process due to the dynamic nature of the conception that is task-dependent and context-dependent. Practical significance of the study is that it helps to recognize misconceptions and starts addressing them so students will have a more comprehensive view of fundamental mathematical ideas as they progress through the Calculus coursework sequence. The developed duality concept development framework called Action-Process-Object-Duality (APOD) adapted from the APOS theory could guide educators and researchers as they engage in assessing students' conception of duality. The results of this study could serve as a facilitating instrument to further analyze cognitive obstacles in college students' understanding of the infinity concept.

Triangles with Integer Side Lengths and Rational Internal Radius P and External Radius R

ERIC Educational Resources Information Center

Zelator, Konstantine

2005-01-01

This paper is written on a level accessible to college/university students of mathematics who are taking second-year, algebra based, mathematics courses beyond calculus I. This article combines material from geometry, trigonometry, and number theory. This integration of various techniques is an excellent experience for the serious student. The…

Approach to Mathematics in Textbooks at Tertiary Level--Exploring Authors' Views about Their Texts

ERIC Educational Resources Information Center

Randahl, Mira

2012-01-01

The aim of this article is to present and discuss some results from an inquiry into mathematics textbooks authors' visions about their texts and approaches they choose when new concepts are introduced. Authors' responses are discussed in relation to results about students' difficulties with approaching calculus reported by previous research. A…

Observations on Student Difficulties with Mathematics in Upper-Division Electricity and Magnetism

ERIC Educational Resources Information Center

Pepper, Rachel E.; Chasteen, Stephanie V.; Pollock, Steven J.; Perkins, Katherine K.

2012-01-01

We discuss common difficulties in upper-division electricity and magnetism (E&M) in the areas of Gauss's law, vector calculus, and electric potential using both quantitative and qualitative evidence. We also show that many of these topical difficulties may be tied to student difficulties with mathematics. At the junior level, some students…

The Use of Visual Approach in Teaching and Learning the Epsilon-Delta Definition of Continuity

ERIC Educational Resources Information Center

Pešic, Duška; Pešic, Aleksandar

2015-01-01

In this paper we introduce a new collaborative technique in teaching and learning the epsilon-delta definition of a continuous function at the point from its domain, which connects mathematical logic, combinatorics and calculus. This collaborative approach provides an opportunity for mathematical high school students to engage in mathematical…

Integration of Digital Technology and Innovative Strategies for Learning and Teaching Large Classes: A Calculus Case Study

ERIC Educational Resources Information Center

Vajravelu, Kuppalapalle; Muhs, Tammy

2016-01-01

Successful science and engineering programs require proficiency and dynamics in mathematics classes to enhance the learning of complex subject matter with a sufficient amount of practical problem solving. Improving student performance and retention in mathematics classes requires inventive approaches. At the University of Central Florida (UCF) the…

Investigating a Link between Pre-Calculus Students' Uses of Graphing Calculators and Their Understanding of Mathematical Symbols

ERIC Educational Resources Information Center

Kenney, Rachael H.

2014-01-01

This study examined ways in which students make use of a graphing calculator and how use relates to comfort and understanding with mathematical symbols. Analysis involved examining students' words and actions in problem solving to identify evidence of algebraic insight. Findings suggest that some symbols and symbolic structures had strong…

Calculus Expectations: Comparisons by High School and College Faculty of What Constitutes Adequate Preparation

ERIC Educational Resources Information Center

Stroumbakis, Konstantinos

2010-01-01

Completion of higher level high school mathematics courses need not translate to success in introductory college level mathematics courses, which, in turn, may contribute to attrition from STEM programs. High school and college faculty rated online survey items, corresponding to content and pedagogy, with respect to importance for success in…

Relationship between High School Mathematical Achievement and Quantitative GPA

ERIC Educational Resources Information Center

Brown, Jennifer L.; Halpin, Glennelle; Halpin, Gerald

2015-01-01

The demand for STEM graduates has increased, but the number of incoming freshmen who declare a STEM major has remained stagnant. High school courses, such as calculus, can open or close the gate for students interested in careers in STEM. The purpose of this study was to determine if high school mathematics preparation was a significant…

The Relationship between Gender and Students' Attitude and Experience of Using a Mathematical Software Program (MATLAB)

ERIC Educational Resources Information Center

Ocak, Mehmet A.

2006-01-01

This correlation study examined the relationship between gender and the students' attitude and prior knowledge of using one of the mathematical software programs (MATLAB). Participants were selected from one community college, one state university and one private college. Students were volunteers from three Calculus I classrooms (one class from…

How Do Adults Perceive, Analyse and Measure Slope?

ERIC Educational Resources Information Center

Duncan, Bruce; Chick, Helen

2013-01-01

Slope is a mathematical concept that is both fundamental to the study of advanced calculus and commonly perceived in everyday life. The measurement of steepness of terrain as a ratio is an example of an everyday application the concept of slope. In this study, a group of pre-service teachers were tested for their capacity to mathematize the…

A Study of Pre-Service Elementary Teachers' Mathematical Sophistication in a Reform-Oriented Calculus Course

ERIC Educational Resources Information Center

Ritter, Carrie Lineberry

2015-01-01

Calls for better preparation of STEM teachers have been prominent in educational communities and among the public for the past several years (e.g. American Association of Colleges for Teacher Education, 2007). Some research suggests one way to improve mathematics instruction is to increase elementary pre-service teachers' "mathematical…

Experience, gender, and performance: Connecting high school physics experience and gender differences to introductory college physics performance

NASA Astrophysics Data System (ADS)

Tai, Robert H.

Current science educational practice is coming under heavy criticism based on the dismaying results of the Third International Mathematics and Science Study of 1998, the latest in a series of large scale surveys; and from research showing the appallingly low representation of females in science-related fields. These critical evaluations serve to draw attention to science literacy in general and lack of persistence among females in particular, two issues that relate closely to the "preparation for future study" goal held by many high school science teachers. In other words, these teachers often seek to promote future success and to prevent future failure in their students' academic careers. This thesis studies the connection between the teaching practices recommended by reformers and researchers for high school teachers, and their students' subsequent college physics performance. The teaching practices studied were: laboratory experiences, class discussion experiences, content coverage, and reliance on textbooks. This study analyzed a survey of 1500 students from 16 different lecture-format college physics courses at 14 different universities. Using hierarchical linear modeling, this study accounted for course-level variables (Calculus-based/Non-calculus course type, professor's gender, and university selectivity). This study controlled for the student's parents education, high school science/mathematics achievement, high school calculus background, and racial background. In addition, the interactions between gender and both pedagogical/curricular and course-level variables were analyzed. The results indicated that teaching fewer topics in greater depth in high school physics appeared to be helpful to college physics students. An interaction between college course type and content coverage showed that students in Calculus-based physics reaped even greater benefits from a depth-oriented curriculum. Also students with fewer labs per month in high school physics appeared to perform better in college physics than did students with many more labs per month. The only significant interaction was between gender and Calculus-based/Non-calculus college course type. Females appeared to do better on average than their males counterparts in Non-calculus physics, but this trend is clearly reversed for Calculus-based physics. This is a disturbing result for educators who have worked to promote persistence among women in engineering and science research. Recommendations are included for high school physics teachers, students and their parents, and college physics instructors.

Combining ultrasonography and noncontrast helical computerized tomography to evaluate Holmium laser lithotripsy

PubMed Central

Mi, Jia; Li, Jie; Zhang, Qinglu; Wang, Xing; Liu, Hongyu; Cao, Yanlu; Liu, Xiaoyan; Sun, Xiao; Shang, Mengmeng; Liu, Qing

2016-01-01

Abstract The purpose of the study was to establish a mathematical model for correlating the combination of ultrasonography and noncontrast helical computerized tomography (NCHCT) with the total energy of Holmium laser lithotripsy. In this study, from March 2013 to February 2014, 180 patients with single urinary calculus were examined using ultrasonography and NCHCT before Holmium laser lithotripsy. The calculus location and size, acoustic shadowing (AS) level, twinkling artifact intensity (TAI), and CT value were all documented. The total energy of lithotripsy (TEL) and the calculus composition were also recorded postoperatively. Data were analyzed using Spearman's rank correlation coefficient, with the SPSS 17.0 software package. Multiple linear regression was also used for further statistical analysis. A significant difference in the TEL was observed between renal calculi and ureteral calculi (r = –0.565, P < 0.001), and there was a strong correlation between the calculus size and the TEL (r = 0.675, P < 0.001). The difference in the TEL between the calculi with and without AS was highly significant (r = 0.325, P < 0.001). The CT value of the calculi was significantly correlated with the TEL (r = 0.386, P < 0.001). A correlation between the TAI and TEL was also observed (r = 0.391, P < 0.001). Multiple linear regression analysis revealed that the location, size, and TAI of the calculi were related to the TEL, and the location and size were statistically significant predictors (adjusted r2 = 0.498, P < 0.001). A mathematical model correlating the combination of ultrasonography and NCHCT with TEL was established; this model may provide a foundation to guide the use of energy in Holmium laser lithotripsy. The TEL can be estimated by the location, size, and TAI of the calculus. PMID:27930563

Mathematical Methods for Physics and Engineering Third Edition Paperback Set

NASA Astrophysics Data System (ADS)

Riley, Ken F.; Hobson, Mike P.; Bence, Stephen J.

2006-06-01

Prefaces; 1. Preliminary algebra; 2. Preliminary calculus; 3. Complex numbers and hyperbolic functions; 4. Series and limits; 5. Partial differentiation; 6. Multiple integrals; 7. Vector algebra; 8. Matrices and vector spaces; 9. Normal modes; 10. Vector calculus; 11. Line, surface and volume integrals; 12. Fourier series; 13. Integral transforms; 14. First-order ordinary differential equations; 15. Higher-order ordinary differential equations; 16. Series solutions of ordinary differential equations; 17. Eigenfunction methods for differential equations; 18. Special functions; 19. Quantum operators; 20. Partial differential equations: general and particular; 21. Partial differential equations: separation of variables; 22. Calculus of variations; 23. Integral equations; 24. Complex variables; 25. Application of complex variables; 26. Tensors; 27. Numerical methods; 28. Group theory; 29. Representation theory; 30. Probability; 31. Statistics; Index.

Fractional calculus in hydrologic modeling: A numerical perspective

PubMed Central

Benson, David A.; Meerschaert, Mark M.; Revielle, Jordan

2013-01-01

Fractional derivatives can be viewed either as handy extensions of classical calculus or, more deeply, as mathematical operators defined by natural phenomena. This follows the view that the diffusion equation is defined as the governing equation of a Brownian motion. In this paper, we emphasize that fractional derivatives come from the governing equations of stable Lévy motion, and that fractional integration is the corresponding inverse operator. Fractional integration, and its multi-dimensional extensions derived in this way, are intimately tied to fractional Brownian (and Lévy) motions and noises. By following these general principles, we discuss the Eulerian and Lagrangian numerical solutions to fractional partial differential equations, and Eulerian methods for stochastic integrals. These numerical approximations illuminate the essential nature of the fractional calculus. PMID:23524449

National Center for Mathematics and Science - K-12 education research

Science.gov Websites

motion, calculus, statistics, genetics, evolution, astronomy, and other topics. Teacher professional ). Extensive materials developed for instruction in evolutionary biology and astronomy - using the model-based

DOE Fundamentals Handbook: Mathematics, Volume 1

DOE Office of Scientific and Technical Information (OSTI.GOV)

Not Available

1992-06-01

The Mathematics Fundamentals Handbook was developed to assist nuclear facility operating contractors provide operators, maintenance personnel, and the technical staff with the necessary fundamentals training to ensure a basic understanding of mathematics and its application to facility operation. The handbook includes a review of introductory mathematics and the concepts and functional use of algebra, geometry, trigonometry, and calculus. Word problems, equations, calculations, and practical exercises that require the use of each of the mathematical concepts are also presented. This information will provide personnel with a foundation for understanding and performing basic mathematical calculations that are associated with various DOE nuclearmore » facility operations.« less

DOE Fundamentals Handbook: Mathematics, Volume 2

DOE Office of Scientific and Technical Information (OSTI.GOV)

Not Available

1992-06-01

The Mathematics Fundamentals Handbook was developed to assist nuclear facility operating contractors provide operators, maintenance personnel, and the technical staff with the necessary fundamentals training to ensure a basic understanding of mathematics and its application to facility operation. The handbook includes a review of introductory mathematics and the concepts and functional use of algebra, geometry, trigonometry, and calculus. Word problems, equations, calculations, and practical exercises that require the use of each of the mathematical concepts are also presented. This information will provide personnel with a foundation for understanding and performing basic mathematical calculations that are associated with various DOE nuclearmore » facility operations.« less

Averages, Areas and Volumes; Cambridge Conference on School Mathematics Feasibility Study No. 45.

ERIC Educational Resources Information Center

Cambridge Conference on School Mathematics, Newton, MA.

Presented is an elementary approach to areas, columns and other mathematical concepts usually treated in calculus. The approach is based on the idea of average and this concept is utilized throughout the report. In the beginning the average (arithmetic mean) of a set of numbers is considered and two properties of the average which often simplify…

Klein's Plan B in the Early Teaching of Analysis: Two Theoretical Cases of Exploring Mathematical Links

ERIC Educational Resources Information Center

Kondratieva, Margo; Winsløw, Carl

2018-01-01

We present a theoretical approach to the problem of the transition from Calculus to Analysis within the undergraduate mathematics curriculum. First, we formulate this problem using the anthropological theory of the didactic, in particular the notion of praxeology, along with a possible solution related to Klein's "Plan B": here,…

Logic for Physicists

NASA Astrophysics Data System (ADS)

Pereyra, Nicolas A.

2018-06-01

This book gives a rigorous yet 'physics-focused' introduction to mathematical logic that is geared towards natural science majors. We present the science major with a robust introduction to logic, focusing on the specific knowledge and skills that will unavoidably be needed in calculus topics and natural science topics in general (rather than taking a philosophical-math-fundamental oriented approach that is commonly found in mathematical logic textbooks).

Does Raising the Bar Level the Playing Field?: Mathematics Curricular Intensification and Inequality in American High Schools, 1982-2004

ERIC Educational Resources Information Center

Domina, Thurston; Saldana, Joshua

2012-01-01

Over the past three decades, American high school students' course taking has rapidly intensified. Between 1982 and 2004, for example, the proportion of high school graduates who earned credit in precalculus or calculus more than tripled. In this article, the authors investigate the consequences of mathematics curricular intensification for social…

Computational tools for Breakthrough Propulsion Physics: State of the art and future prospects

NASA Astrophysics Data System (ADS)

Maccone, Claudio

2000-01-01

To address problems in Breakthrough Propulsion Physics (BPP) one needs sheer computing capabilities. This is because General Relativity and Quantum Field Theory are so mathematically sophisticated that the amount of analytical calculations is prohibitive and one can hardly do all of them by hand. In this paper we make a comparative review of the main tensor calculus capabilities of the three most advanced and commercially available ``symbolic manipulator'' codes: Macsyma, Maple V and Mathematica. We also point out that currently one faces such a variety of different conventions in tensor calculus that it is difficult or impossible to compare results obtained by different scholars in General Relativity and Quantum Field Theory. Mathematical physicists, experimental physicists and engineers have each their own way of customizing tensors, especially by using the different metric signatures, different metric determinant signs, different definitions of the basic Riemann and Ricci tensors, and by adopting different systems of physical units. This chaos greatly hampers progress toward the chief NASA BPP goal: the design of the NASA Warp Drive. It is thus concluded that NASA should put order by establishing international standards in symbolic tensor calculus and enforcing anyone working in BPP to adopt these NASA BPP Standards. .

Software Reviews.

ERIC Educational Resources Information Center

Bitter, Gary G., Ed.

1989-01-01

Describes three software packages: (1) "MacMendeleev"--database/graphic display for chemistry, grades 10-12, Macintosh; (2) "Geometry One: Foundations"--geometry tutorial, grades 7-12, IBM; (3) "Mathematics Exploration Toolkit"--algebra and calculus tutorial, grades 8-12, IBM. (MVL)

Analysis of the coupling efficiency of a tapered space receiver with a calculus mathematical model

NASA Astrophysics Data System (ADS)

Hu, Qinggui; Mu, Yining

2018-03-01

We establish a calculus mathematical model to study the coupling characteristics of tapered optical fibers in a space communications system, and obtained the coupling efficiency equation. Then, using MATLAB software, the solution was calculated. After this, the sample was produced by the mature flame-brush technique. The experiment was then performed, and the results were in accordance with the theoretical analysis. This shows that the theoretical analysis was correct and indicates that a tapered structure could improve its tolerance with misalignment. Project supported by The National Natural Science Foundation of China (grant no. 61275080); 2017 Jilin Province Science and Technology Development Plan-Science and Technology Innovation Fund for Small and Medium Enterprises (20170308029HJ); ‘thirteen five’ science and technology research project of the Department of Education of Jilin 2016 (16JK009).

A quantitative analysis of the relationship between an online homework system and student achievement in pre-calculus

NASA Astrophysics Data System (ADS)

Babaali, Parisa; Gonzalez, Lidia

2015-07-01

Supporting student success in entry-level mathematics courses at the undergraduate level has and continues to be a challenge. Recently we have seen an increased reliance on technological supports including software to supplement more traditional in-class instruction. In this paper, we explore the effects on student performance of the use of a computer software program to supplement instruction in an entry-level mathematics course at the undergraduate level, specifically, a pre-calculus course. Relying on data from multiple sections of the course over various semesters, we compare student performance in those classes utilizing the software against those in which it was not used. Quantitative analysis of the data then leads us to conclusions about the effectiveness of the software as well as recommendations for future iterations of the course and others like it.

An Introduction to Kristof's Theorem for Solving Least-Square Optimization Problems Without Calculus.

PubMed

Waller, Niels

2018-01-01

Kristof's Theorem (Kristof, 1970 ) describes a matrix trace inequality that can be used to solve a wide-class of least-square optimization problems without calculus. Considering its generality, it is surprising that Kristof's Theorem is rarely used in statistics and psychometric applications. The underutilization of this method likely stems, in part, from the mathematical complexity of Kristof's ( 1964 , 1970 ) writings. In this article, I describe the underlying logic of Kristof's Theorem in simple terms by reviewing four key mathematical ideas that are used in the theorem's proof. I then show how Kristof's Theorem can be used to provide novel derivations to two cognate models from statistics and psychometrics. This tutorial includes a glossary of technical terms and an online supplement with R (R Core Team, 2017 ) code to perform the calculations described in the text.

Mathematical Minute: Rotating a Function Graph

ERIC Educational Resources Information Center

Bravo, Daniel; Fera, Joseph

2013-01-01

Using calculus only, we find the angles you can rotate the graph of a differentiable function about the origin and still obtain a function graph. We then apply the solution to odd and even degree polynomials.

Software Reviews.

ERIC Educational Resources Information Center

Mathematics and Computer Education, 1987

1987-01-01

Presented are reviews of several microcomputer software programs. Included are reviews of: (1) Microstat (Zenith); (2) MathCAD (MathSoft); (3) Discrete Mathematics (True Basic); (4) CALCULUS (True Basic); (5) Linear-Kit (John Wiley); and (6) Geometry Sensei (Broderbund). (RH)

Mathematics Conceptual Visualization with HyperCard.

ERIC Educational Resources Information Center

Haws, LaDawn

1992-01-01

Hypermedia provides an easy-to-use option for adding visualization, via the computer, to the classroom. Some examples of this medium are presented, including applications in basic linear algebra and calculus, and a tutorial in electromagnetism. (Author)

Visualizing Volume to Help Students Understand the Disk Method on Calculus Integral Course

NASA Astrophysics Data System (ADS)

Tasman, F.; Ahmad, D.

2018-04-01

Many research shown that students have difficulty in understanding the concepts of integral calculus. Therefore this research is interested in designing a classroom activity integrated with design research method to assist students in understanding the integrals concept especially in calculating the volume of rotary objects using disc method. In order to support student development in understanding integral concepts, this research tries to use realistic mathematical approach by integrating geogebra software. First year university student who takes a calculus course (approximately 30 people) was chosen to implement the classroom activity that has been designed. The results of retrospective analysis show that visualizing volume of rotary objects using geogebra software can assist the student in understanding the disc method as one way of calculating the volume of a rotary object.

Student Solution Manual for Mathematical Methods for Physics and Engineering Third Edition

NASA Astrophysics Data System (ADS)

Riley, K. F.; Hobson, M. P.

2006-03-01

Preface; 1. Preliminary algebra; 2. Preliminary calculus; 3. Complex numbers and hyperbolic functions; 4. Series and limits; 5. Partial differentiation; 6. Multiple integrals; 7. Vector algebra; 8. Matrices and vector spaces; 9. Normal modes; 10. Vector calculus; 11. Line, surface and volume integrals; 12. Fourier series; 13. Integral transforms; 14. First-order ordinary differential equations; 15. Higher-order ordinary differential equations; 16. Series solutions of ordinary differential equations; 17. Eigenfunction methods for differential equations; 18. Special functions; 19. Quantum operators; 20. Partial differential equations: general and particular; 21. Partial differential equations: separation of variables; 22. Calculus of variations; 23. Integral equations; 24. Complex variables; 25. Application of complex variables; 26. Tensors; 27. Numerical methods; 28. Group theory; 29. Representation theory; 30. Probability; 31. Statistics.

In Praise of the Catenary

NASA Astrophysics Data System (ADS)

Behroozi, F.

2018-04-01

When a chain hangs loosely from its end points, it takes the familiar form known as the catenary. Power lines, clothes lines, and chain links are familiar examples of the catenary in everyday life. Nevertheless, the subject is conspicuously absent from current introductory physics and calculus courses. Even in upper-level physics and math courses, the catenary equation is usually introduced as an example of hyperbolic functions or discussed as an application of the calculus of variations. We present a new derivation of the catenary equation that is suitable for introductory physics and mathematics courses.

Geometry and physics of pseudodifferential operators on manifolds

NASA Astrophysics Data System (ADS)

Esposito, Giampiero; Napolitano, George M.

2016-09-01

A review is made of the basic tools used in mathematics to define a calculus for pseudodifferential operators on Riemannian manifolds endowed with a connection: existence theorem for the function that generalizes the phase; analogue of Taylor's theorem; torsion and curvature terms in the symbolic calculus; the two kinds of derivative acting on smooth sections of the cotangent bundle of the Riemannian manifold; the concept of symbol as an equivalence class. Physical motivations and applications are then outlined, with emphasis on Green functions of quantum field theory and Parker's evaluation of Hawking radiation.

Curve Fitting via the Criterion of Least Squares. Applications of Algebra and Elementary Calculus to Curve Fitting. [and] Linear Programming in Two Dimensions: I. Applications of High School Algebra to Operations Research. Modules and Monographs in Undergraduate Mathematics and Its Applications Project. UMAP Units 321, 453.

ERIC Educational Resources Information Center

Alexander, John W., Jr.; Rosenberg, Nancy S.

This document consists of two modules. The first of these views applications of algebra and elementary calculus to curve fitting. The user is provided with information on how to: 1) construct scatter diagrams; 2) choose an appropriate function to fit specific data; 3) understand the underlying theory of least squares; 4) use a computer program to…

Proceedings of the Conference of the International Group for the Psychology of Mathematics Education (21st, Lahti, Finland, July 14-19, 1997). Volume 3.

ERIC Educational Resources Information Center

Pehkonen, Erkki, Ed.

The third volume of the proceedings of 21st annual meeting of the International Group for the Psychology of Mathematics Education contains the following papers: (1) "Graphics Calculators Use in Precalculus and Achievement in Calculus" (P. Gomez and F. Femandez); (2) "Tapping into Algebraic Variables through the Graphic…

On the use of history of mathematics: an introduction to Galileo's study of free fall motion

NASA Astrophysics Data System (ADS)

Ponce Campuzano, Juan Carlos; Matthews, Kelly E.; Adams, Peter

2018-05-01

In this paper, we report on an experimental activity for discussing the concepts of speed, instantaneous speed and acceleration, generally introduced in first year university courses of calculus or physics. Rather than developing the ideas of calculus and using them to explain these basic concepts for the study of motion, we led 82 first year university students through Galileo's experiments designed to investigate the motion of falling bodies, and his geometrical explanation of his results, via simple dynamic geometric applets designed with GeoGebra. Our goal was to enhance the students' development of mathematical thinking. Through a scholarship of teaching and learning study design, we captured data from students before, during and after the activity. Findings suggest that the historical development presented to the students helped to show the growth and evolution of the ideas and made visible authentic ways of thinking mathematically. Importantly, the activity prompted students to question and rethink what they knew about speed and acceleration, and also to appreciate the novel concepts of instantaneous speed and acceleration at which Galileo arrived.

Fractional calculus in bioengineering, part 3.

PubMed

Magin, Richard L

2004-01-01

Fractional calculus (integral and differential operations of noninteger order) is not often used to model biological systems. Although the basic mathematical ideas were developed long ago by the mathematicians Leibniz (1695), Liouville (1834), Riemann (1892), and others and brought to the attention of the engineering world by Oliver Heaviside in the 1890s, it was not until 1974 that the first book on the topic was published by Oldham and Spanier. Recent monographs and symposia proceedings have highlighted the application of fractional calculus in physics, continuum mechanics, signal processing, and electromagnetics, but with few examples of applications in bioengineering. This is surprising because the methods of fractional calculus, when defined as a Laplace or Fourier convolution product, are suitable for solving many problems in biomedical research. For example, early studies by Cole (1933) and Hodgkin (1946) of the electrical properties of nerve cell membranes and the propagation of electrical signals are well characterized by differential equations of fractional order. The solution involves a generalization of the exponential function to the Mittag-Leffler function, which provides a better fit to the observed cell membrane data. A parallel application of fractional derivatives to viscoelastic materials establishes, in a natural way, hereditary integrals and the power law (Nutting/Scott Blair) stress-strain relationship for modeling biomaterials. In this review, I will introduce the idea of fractional operations by following the original approach of Heaviside, demonstrate the basic operations of fractional calculus on well-behaved functions (step, ramp, pulse, sinusoid) of engineering interest, and give specific examples from electrochemistry, physics, bioengineering, and biophysics. The fractional derivative accurately describes natural phenomena that occur in such common engineering problems as heat transfer, electrode/electrolyte behavior, and sub-threshold nerve propagation. By expanding the range of mathematical operations to include fractional calculus, we can develop new and potentially useful functional relationships for modeling complex biological systems in a direct and rigorous manner. In Part 2 of this review (Crit Rev Biomed Eng 2004; 32(1):105-193), fractional calculus was applied to problems in nerve stimulation, dielectric relaxation, and viscoelastic materials by extending the governing differential equations to include fractional order terms. In this third and final installment, we consider distributed systems that represent shear stress in fluids, heat transfer in uniform one-dimensional media, and subthreshold nerve depolarization. Classic electrochemical analysis and impedance spectroscopy are also reviewed from the perspective of fractional calculus, and selected examples from recent studies in neuroscience, bioelectricity, and tissue biomechanics are analyzed to illustrate the vitality of the field.

Reflective Properties of a Parabolic Mirror.

ERIC Educational Resources Information Center

Ramsey, Gordon P.

1991-01-01

An incident light ray parallel to the optical axis of a parabolic mirror will be reflected at the focal point and vice versa. Presents a mathematical proof that uses calculus, algebra, and geometry to prove this reflective property. (MDH)

The Product and Quotient Rules Revisited

ERIC Educational Resources Information Center

Eggleton, Roger; Kustov, Vladimir

2011-01-01

Mathematical elegance is illustrated by strikingly parallel versions of the product and quotient rules of basic calculus, with some applications. Corresponding rules for second derivatives are given: the product rule is familiar, but the quotient rule is less so.

Computer Tensor Codes to Design the War Drive

NASA Astrophysics Data System (ADS)

Maccone, C.

To address problems in Breakthrough Propulsion Physics (BPP) and design the Warp Drive one needs sheer computing capabilities. This is because General Relativity (GR) and Quantum Field Theory (QFT) are so mathematically sophisticated that the amount of analytical calculations is prohibitive and one can hardly do all of them by hand. In this paper we make a comparative review of the main tensor calculus capabilities of the three most advanced and commercially available “symbolic manipulator” codes. We also point out that currently one faces such a variety of different conventions in tensor calculus that it is difficult or impossible to compare results obtained by different scholars in GR and QFT. Mathematical physicists, experimental physicists and engineers have each their own way of customizing tensors, especially by using different metric signatures, different metric determinant signs, different definitions of the basic Riemann and Ricci tensors, and by adopting different systems of physical units. This chaos greatly hampers progress toward the design of the Warp Drive. It is thus suggested that NASA would be a suitable organization to establish standards in symbolic tensor calculus and anyone working in BPP should adopt these standards. Alternatively other institutions, like CERN in Europe, might consider the challenge of starting the preliminary implementation of a Universal Tensor Code to design the Warp Drive.

Exploring K-12 mathematics course progression: implications for collegiate success in Florida

NASA Astrophysics Data System (ADS)

Campbell, Bethany; Varney, Christopher; Wade, Aaron

Increasingly, Florida college students are pressured to change their major as few times as possible and take only required classes, all in order to ``Finish in Four, Save More''. If they fail to do so, they may be subject to penalties such as Excess Hour Fees. Partially as a result of this, students wishing to study STEM are at a significant disadvantage if they enter college unprepared to take calculus their first semester. We explore the various ``paths to success'' to STEM degrees, defined by entering college having taken calculus in high school, starting from fifth grade onwards.

An accessible four-dimensional treatment of Maxwell's equations in terms of differential forms

NASA Astrophysics Data System (ADS)

Sá, Lucas

2017-03-01

Maxwell’s equations are derived in terms of differential forms in the four-dimensional Minkowski representation, starting from the three-dimensional vector calculus differential version of these equations. Introducing all the mathematical and physical concepts needed (including the tool of differential forms), using only knowledge of elementary vector calculus and the local vector version of Maxwell’s equations, the equations are reduced to a simple and elegant set of two equations for a unified quantity, the electromagnetic field. The treatment should be accessible for students taking a first course on electromagnetism.

Jan Hudde and the Quotient Rule before Newton and Leibniz

ERIC Educational Resources Information Center

Curtin, Daniel J.

2005-01-01

This article describes some of the work of Jan Hudde who anticipated some results of calculus. Prior to a career as a Burgomaster of Amsterdam, Hudde engaged in mathematics. His method of finding maxima and minima is especially interesting.

A Rolling Sphere on a Tilted Rotating Turntable.

ERIC Educational Resources Information Center

Sambles, J. R.; And Others

1983-01-01

Describes an advanced high school/college experiment that illustrates the mechanics describing the motion of a rolling ball. Includes procedures used, discussions of vectoral and mathematical (calculus) solutions to the investigation, and sample student results using the recommended materials. (JM)

Proceedings of the Conference of the International Group for the Psychology of Mathematics Education (PME 20) (20th, Valencia, Spain, July 8-12, 1996). Volume 2.

ERIC Educational Resources Information Center

Puig, Luis, Ed.; Gutierrez, Angel, Ed.

The second volume of this proceedings contains full research articles. Papers include: (1) "Lave and Wenger's social practice theory and teaching and learning school mathematics" (J. Adler); (2) "Being a researcher and being a teacher" (J. Ainley); (3) "Procedural and conceptual aspects of standard algorithms in calculus" (M.B. Ali and D. Tall);…

Critical Analysis of the Mathematical Formalism of Theoretical Physics. I. Foundations of Differential and Integral Calculus

NASA Astrophysics Data System (ADS)

Kalanov, Temur Z.

2013-04-01

Critical analysis of the standard foundations of differential and integral calculus -- as mathematical formalism of theoretical physics -- is proposed. Methodological basis of the analysis is the unity of formal logic and rational dialectics. It is shown that: (a) the foundations (i.e. d 1ptyd,;=;δ,;->;0,;δ,δ,, δ,;->;0;δ,δ,;=;δ,;->;0;f,( x;+;δ, );-;f,( x )δ,;, d,;=;δ,, d,;=;δ, where y;=;f,( x ) is a continuous function of one argument x; δ, and δ, are increments; d, and d, are differentials) not satisfy formal logic law -- the law of identity; (b) the infinitesimal quantities d,, d, are fictitious quantities. They have neither algebraic meaning, nor geometrical meaning because these quantities do not take numerical values and, therefore, have no a quantitative measure; (c) expressions of the kind x;+;d, are erroneous because x (i.e. finite quantity) and d, (i.e. infinitely diminished quantity) have different sense, different qualitative determinacy; since x;,;,,,,onst under δ,;,;,, a derivative does not contain variable quantity x and depends only on constant c. Consequently, the standard concepts ``infinitesimal quantity (uninterruptedly diminishing quantity)'', ``derivative'', ``derivative as function of variable quantity'' represent incorrect basis of mathematics and theoretical physics.

Condition-based diagnosis of mechatronic systems using a fractional calculus approach

NASA Astrophysics Data System (ADS)

Gutiérrez-Carvajal, Ricardo Enrique; Flávio de Melo, Leonimer; Maurício Rosário, João; Tenreiro Machado, J. A.

2016-07-01

While fractional calculus (FC) is as old as integer calculus, its application has been mainly restricted to mathematics. However, many real systems are better described using FC equations than with integer models. FC is a suitable tool for describing systems characterised by their fractal nature, long-term memory and chaotic behaviour. It is a promising methodology for failure analysis and modelling, since the behaviour of a failing system depends on factors that increase the model's complexity. This paper explores the proficiency of FC in modelling complex behaviour by tuning only a few parameters. This work proposes a novel two-step strategy for diagnosis, first modelling common failure conditions and, second, by comparing these models with real machine signals and using the difference to feed a computational classifier. Our proposal is validated using an electrical motor coupled with a mechanical gear reducer.

Wave propagation in viscoelastic horns using a fractional calculus rheology model

NASA Astrophysics Data System (ADS)

Margulies, Timothy

2003-10-01

The complex mechanical behavior of materials are characterized by fluid and solid models with fractional calculus differentials to relate stress and strain fields. Fractional derivatives have been shown to describe the viscoelastic stress from polymer chain theory for molecular solutions [Rouse and Sittel, J. Appl. Phys. 24, 690 (1953)]. Here the propagation of infinitesimal waves in one dimensional horns with a small cross-sectional area change along the longitudinal axis are examined. In particular, the linear, conical, exponential, and catenoidal shapes are studied. The wave amplitudes versus frequency are solved analytically and predicted with mathematical computation. Fractional rheology data from Bagley [J. Rheol. 27, 201 (1983); Bagley and Torvik, J. Rheol. 30, 133 (1986)] are incorporated in the simulations. Classical elastic and fluid ``Webster equations'' are recovered in the appropriate limits. Horns with real materials that employ fractional calculus representations can be modeled to examine design trade-offs for engineering or for scientific application.

A new collection of real world applications of fractional calculus in science and engineering

NASA Astrophysics Data System (ADS)

Sun, HongGuang; Zhang, Yong; Baleanu, Dumitru; Chen, Wen; Chen, YangQuan

2018-11-01

Fractional calculus is at this stage an arena where many models are still to be introduced, discussed and applied to real world applications in many branches of science and engineering where nonlocality plays a crucial role. Although researchers have already reported many excellent results in several seminal monographs and review articles, there are still a large number of non-local phenomena unexplored and waiting to be discovered. Therefore, year by year, we can discover new aspects of the fractional modeling and applications. This review article aims to present some short summaries written by distinguished researchers in the field of fractional calculus. We believe this incomplete, but important, information will guide young researchers and help newcomers to see some of the main real-world applications and gain an understanding of this powerful mathematical tool. We expect this collection will also benefit our community.

Related Rates and the Speed of Light.

ERIC Educational Resources Information Center

Althoen, S. C.; Weidner, J. F.

1985-01-01

Standard calculus textbooks often include a related rates problem involving light cast onto a straight line by a revolving light source. Mathematical aspects to these problems (both in the solution and in the method by which that solution is obtained) are examined. (JN)

Integrating Security into the Curriculum

DTIC Science & Technology

1998-12-01

predicate calculus, discrete math , and finite-state machine the- ory. In addition to applying standard mathematical foundations to constructing hardware and...models, specifi- cations, and the use of formal methods for verification and covert channel analysis. The means for analysis is based on discrete math , information

On Solving Linear Recurrences

ERIC Educational Resources Information Center

Dobbs, David E.

2013-01-01

A direct method is given for solving first-order linear recurrences with constant coefficients. The limiting value of that solution is studied as "n to infinity." This classroom note could serve as enrichment material for the typical introductory course on discrete mathematics that follows a calculus course.

Continuous Problem of Function Continuity

ERIC Educational Resources Information Center

Jayakody, Gaya; Zazkis, Rina

2015-01-01

We examine different definitions presented in textbooks and other mathematical sources for "continuity of a function at a point" and "continuous function" in the context of introductory level Calculus. We then identify problematic issues related to definitions of continuity and discontinuity: inconsistency and absence of…

Integrating External Software into SMART Board™ Calculus Lessons

ERIC Educational Resources Information Center

Wolmer, Allen; Khazanov, Leonid

2011-01-01

Interactive Whiteboards (IWBs) are becoming commonplace throughout primary, secondary, and postsecondary classrooms. However, the focus of the associated lesson creation & management software tools delivered with IWBs has been the primary grades, while secondary and postsecondary mathematics lessons have requirements beyond what is delivered…

Subject design and factors affecting achievement in mathematics for biomedical science

NASA Astrophysics Data System (ADS)

Carnie, Steven; Morphett, Anthony

2017-01-01

Reports such as Bio2010 emphasize the importance of integrating mathematical modelling skills into undergraduate biology and life science programmes, to ensure students have the skills and knowledge needed for biological research in the twenty-first century. One way to do this is by developing a dedicated mathematics subject to teach modelling and mathematical concepts in biological contexts. We describe such a subject at a research-intensive Australian university, and discuss the considerations informing its design. We also present an investigation into the effect of mathematical and biological background, prior mathematical achievement, and gender, on student achievement in the subject. The investigation shows that several factors known to predict performance in standard calculus subjects apply also to specialized discipline-specific mathematics subjects, and give some insight into the relative importance of mathematical versus biological background for a biology-focused mathematics subject.

Establishing an Explanatory Model for Mathematics Identity.

PubMed

Cribbs, Jennifer D; Hazari, Zahra; Sonnert, Gerhard; Sadler, Philip M

2015-04-01

This article empirically tests a previously developed theoretical framework for mathematics identity based on students' beliefs. The study employs data from more than 9,000 college calculus students across the United States to build a robust structural equation model. While it is generally thought that students' beliefs about their own competence in mathematics directly impact their identity as a "math person," findings indicate that students' self-perceptions related to competence and performance have an indirect effect on their mathematics identity, primarily by association with students' interest and external recognition in mathematics. Thus, the model indicates that students' competence and performance beliefs are not sufficient for their mathematics identity development, and it highlights the roles of interest and recognition. © 2015 The Authors. Child Development © 2015 Society for Research in Child Development, Inc.

Introductory life science mathematics and quantitative neuroscience courses.

PubMed

Duffus, Dwight; Olifer, Andrei

2010-01-01

We describe two sets of courses designed to enhance the mathematical, statistical, and computational training of life science undergraduates at Emory College. The first course is an introductory sequence in differential and integral calculus, modeling with differential equations, probability, and inferential statistics. The second is an upper-division course in computational neuroscience. We provide a description of each course, detailed syllabi, examples of content, and a brief discussion of the main issues encountered in developing and offering the courses.

Metaphorical motion in mathematical reasoning: further evidence for pre-motor implementation of structure mapping in abstract domains.

PubMed

Fields, Chris

2013-08-01

The theory of computation and category theory both employ arrow-based notations that suggest that the basic metaphor "state changes are like motions" plays a fundamental role in all mathematical reasoning involving formal manipulations. If this is correct, structure-mapping inferences implemented by the pre-motor action planning system can be expected to be involved in solving any mathematics problems not solvable by table lookups and number line manipulations alone. Available functional imaging studies of multi-digit arithmetic, algebra, geometry and calculus problem solving are consistent with this expectation.

Optimal Assignment Problem Applications of Finite Mathematics to Business and Economics. [and] Difference Equations with Applications. Applications of Difference Equations to Economics and Social Sciences. [and] Selected Applications of Mathematics to Finance and Investment. Applications of Elementary Algebra to Finance. [and] Force of Interest. Applications of Calculus to Finance. UMAP Units 317, 322, 381, 382.

ERIC Educational Resources Information Center

Gale, David; And Others

Four units make up the contents of this document. The first examines applications of finite mathematics to business and economies. The user is expected to learn the method of optimization in optimal assignment problems. The second module presents applications of difference equations to economics and social sciences, and shows how to: 1) interpret…

Diversions: Hilbert and Sierpinski Space-Filling Curves, and beyond

ERIC Educational Resources Information Center

Gough, John

2012-01-01

Space-filling curves are related to fractals, in that they have self-similar patterns. Such space-filling curves were originally developed as conceptual mathematical "monsters", counter-examples to Weierstrassian and Reimannian treatments of calculus and continuity. These were curves that were everywhere-connected but…

Conditional Independence in Applied Probability.

ERIC Educational Resources Information Center

Pfeiffer, Paul E.

This material assumes the user has the background provided by a good undergraduate course in applied probability. It is felt that introductory courses in calculus, linear algebra, and perhaps some differential equations should provide the requisite experience and proficiency with mathematical concepts, notation, and argument. The document is…

Traffic Flow Estimates.

ERIC Educational Resources Information Center

Hart, Vincent G.

1981-01-01

Two examples are given of ways traffic engineers estimate traffic flow. The first, Floating Car Method, involves some basic ideas and the notion of relative velocity. The second, Maximum Traffic Flow, is viewed to involve simple applications of calculus. The material provides insight into specialized applications of mathematics. (MP)

The Legacy of Leonhard Euler--A Tricentennial Tribute

ERIC Educational Resources Information Center

Debnath, Lokenath

2009-01-01

This tricentennial tribute commemorates Euler's major contributions to mathematical and physical sciences. A brief biographical sketch is presented with his major contributions to certain selected areas of number theory, differential and integral calculus, differential equations, solid and fluid mechanics, topology and graph theory, infinite…

Free Software and Multivariable Calculus

ERIC Educational Resources Information Center

Nord, Gail M.

2011-01-01

Calculators and computers make new modes of instruction possible; yet, at the same time they pose hardships for school districts and mathematics educators trying to incorporate technology with limited monetary resources. In the "Standards," a recommended classroom is one in which calculators, computers, courseware, and manipulative materials are…

Minority Engineering Program Pipeline: A Proposal to Increase Minority Student Enrollment and Retention in Engineering

NASA Technical Reports Server (NTRS)

Charity, Pamela C.; Klein, Paul B.; Wadhwa, Bhushan

1995-01-01

The Cleveland State University Minority Engineering Program Pipeline consist of programs which foster engineering career awareness, academic enrichment, and professional development for historically underrepresented minority studies. The programs involved are the Access to Careers in Engineering (ACE) Program for high school pre-engineering students: the LINK Program for undergraduate students pursuing degree which include engineering; and the PEP (Pre-calculus Enrichment Program) and EPIC (Enrichment Program in Calculus) mathematics programs for undergraduate academic enrichment. The pipeline is such that high school graduates from the ACE Program who enroll at Cleveland State University in pursuit of engineering degrees are admitted to the LINK Program for undergraduate level support. LINK Program students are among the minority participants who receive mathematics enrichment through the PEP and EPIC Programs for successful completion of their engineering required math courses. THese programs are interdependent and share the goal of preparing minority students for engineering careers by enabling them to achieve academically and obtain college degree and career related experience.

Are there common mathematical structures in economics and physics?

NASA Astrophysics Data System (ADS)

Mimkes, Jürgen

2016-12-01

Economics is a field that looks into the future. We may know a few things ahead (ex ante), but most things we only know, afterwards (ex post). How can we work in a field, where much of the important information is missing? Mathematics gives two answers: 1. Probability theory leads to microeconomics: the Lagrange function optimizes utility under constraints of economic terms (like costs). The utility function is the entropy, the logarithm of probability. The optimal result is given by a probability distribution and an integrating factor. 2. Calculus leads to macroeconomics: In economics we have two production factors, capital and labour. This requires two dimensional calculus with exact and not-exact differentials, which represent the "ex ante" and "ex post" terms of economics. An integrating factor turns a not-exact term (like income) into an exact term (entropy, the natural production function). The integrating factor is the same as in microeconomics and turns the not-exact field of economics into an exact physical science.

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

NASA Astrophysics Data System (ADS)

Konstantinou-Katzi, Panagiota; Tsolaki, Eleni; Meletiou-Mavrotheris, Maria; Koutselini, Mary

2013-04-01

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.

Introducing computational thinking through hands-on projects using R with applications to calculus, probability and data analysis

NASA Astrophysics Data System (ADS)

Benakli, Nadia; Kostadinov, Boyan; Satyanarayana, Ashwin; Singh, Satyanand

2017-04-01

The goal of this paper is to promote computational thinking among mathematics, engineering, science and technology students, through hands-on computer experiments. These activities have the potential to empower students to learn, create and invent with technology, and they engage computational thinking through simulations, visualizations and data analysis. We present nine computer experiments and suggest a few more, with applications to calculus, probability and data analysis, which engage computational thinking through simulations, visualizations and data analysis. We are using the free (open-source) statistical programming language R. Our goal is to give a taste of what R offers rather than to present a comprehensive tutorial on the R language. In our experience, these kinds of interactive computer activities can be easily integrated into a smart classroom. Furthermore, these activities do tend to keep students motivated and actively engaged in the process of learning, problem solving and developing a better intuition for understanding complex mathematical concepts.

Real-Time Exponential Curve Fits Using Discrete Calculus

NASA Technical Reports Server (NTRS)

Rowe, Geoffrey

2010-01-01

An improved solution for curve fitting data to an exponential equation (y = Ae(exp Bt) + C) has been developed. This improvement is in four areas -- speed, stability, determinant processing time, and the removal of limits. The solution presented avoids iterative techniques and their stability errors by using three mathematical ideas: discrete calculus, a special relationship (be tween exponential curves and the Mean Value Theorem for Derivatives), and a simple linear curve fit algorithm. This method can also be applied to fitting data to the general power law equation y = Ax(exp B) + C and the general geometric growth equation y = Ak(exp Bt) + C.

Care and Feeding of a Paperless, Calculus-based Physics Course

NASA Astrophysics Data System (ADS)

Moore, Christopher; Fuller, Robert; Plano-Clark, Vicki L.; Dunbar, Steven R.

1997-04-01

Technology is playing an increasing role in our lives at home, at work, and in the classroom. We have begun a calculus-based introductory physics course to integrate mathematics and multimedia with the traditional physics content. This course relies on the use of technology to teach physics. We formulated the following rule for the conduct of the course: ''No paper is transferred between instructional staff and students that contains course information or assignments for grading.'' Implementing and maintaining this physics course within the context of the instructor goals will be discussed. Preliminary results of feedback from the students and an evaluation team will be presented.

Computational approach to Thornley's problem by bivariate operational calculus

NASA Astrophysics Data System (ADS)

Bazhlekova, E.; Dimovski, I.

2012-10-01

Thornley's problem is an initial-boundary value problem with a nonlocal boundary condition for linear onedimensional reaction-diffusion equation, used as a mathematical model of spiral phyllotaxis in botany. Applying a bivariate operational calculus we find explicit representation of the solution, containing two convolution products of special solutions and the arbitrary initial and boundary functions. We use a non-classical convolution with respect to the space variable, extending in this way the classical Duhamel principle. The special solutions involved are represented in the form of fast convergent series. Numerical examples are considered to show the application of the present technique and to analyze the character of the solution.

Examining Graphing Calculator Affordances in Learning Pre-Calculus among Undergraduate Students

ERIC Educational Resources Information Center

Nzuki, Francis

2016-01-01

This study examines graphing calculator affordances in learning mathematics among college precalculus students. The study draws from the Cognitive Load Theory (CLT) and the "Intelligent Technology" theoretical framework proposed by Salomon, Perkins, and Globerson (1991). From these perspectives the effects "with" the graphing…

A Glossary for Pre-Calculus

ERIC Educational Resources Information Center

Arnold, Bruce; Kracht, Brenda; Ross, Judy; Teegarden, Terrie; Tompkins, Maurice

2012-01-01

In the deconstruction of the California state standards for trigonometry, linear algebra and mathematical analysis for the Cal-PASS (California Partnership for Achieving Student Success) Content Standards Deconstruction projects, it became apparent that terms were used for which no definition was given. The San Diego Central Cal-PASS Math…

Nonlinear Stochastic Markov Processes and Modeling Uncertainty in Populations

DTIC Science & Technology

2011-07-06

219–232. [26] I. Karatzas and S.E. Shreve, Brownian Motion and Stochastic Calculus, Second Edition, Springer, New York, 1991. [27] F. Klebaner...ubiquitous in mathematics and physics (e.g., particle transport, filtering), biology (population models), finance (e.g., Black-Scholes equations) among other

Calculus of Elementary Functions, Part I. Teacher's Commentary. Revised Edition.

ERIC Educational Resources Information Center

Herriot, Sarah T.; And Others

This course is intended for students who have a thorough knowledge of college preparatory mathematics including algebra, axiomatic geometry, trigonometry, and analytic geometry. It does not assume they have acquired a background of elementary functions. This teacher's guide contains background information, suggested instructional procedures, and…

Math 3011--College Algebra and Trigonometry. Course Outline.

ERIC Educational Resources Information Center

New York Inst. of Tech., Old Westbury.

This document contains the course syllabus and 12 independent practice modules for a college level mathematics course designed to provide the necessary foundation for success in calculus, develop logical thinking skills, and enhance analytic skills through problem solving. Topics include relations and functions; inequalities; complex numbers;…

Math 3320--Technical Mathematics II.

ERIC Educational Resources Information Center

New York Inst. of Tech., Old Westbury.

This document contains the course syllabus and 12 independent practice modules for a college pre-calculus course designed as the second course in a two-semester sequence for students in a Bachelor of Technology program. The course emphasizes applications from technology and verbal problems. Topics include trigonometric functions; identities;…

The conceptual basis of mathematics in cardiology III: linear systems theory and integral transforms.

PubMed

Bates, Jason H T; Sobel, Burton E

2003-05-01

This is the third in a series of four articles developed for the readers of Coronary Artery Disease. Without language ideas cannot be articulated. What may not be so immediately obvious is that they cannot be formulated either. One of the essential languages of cardiology is mathematics. Unfortunately, medical education does not emphasize, and in fact, often neglects empowering physicians to think mathematically. Reference to statistics, conditional probability, multicompartmental modeling, algebra, calculus and transforms is common but often without provision of genuine conceptual understanding. At the University of Vermont College of Medicine, Professor Bates developed a course designed to address these deficiencies. The course covered mathematical principles pertinent to clinical cardiovascular and pulmonary medicine and research. It focused on fundamental concepts to facilitate formulation and grasp of ideas.This series of four articles was developed to make the material available for a wider audience. The articles will be published sequentially in Coronary Artery Disease. Beginning with fundamental axioms and basic algebraic manipulations they address algebra, function and graph theory, real and complex numbers, calculus and differential equations, mathematical modeling, linear system theory and integral transforms and statistical theory. The principles and concepts they address provide the foundation needed for in-depth study of any of these topics. Perhaps of even more importance, they should empower cardiologists and cardiovascular researchers to utilize the language of mathematics in assessing the phenomena of immediate pertinence to diagnosis, pathophysiology and therapeutics. The presentations are interposed with queries (by Coronary Artery Disease abbreviated as CAD) simulating the nature of interactions that occurred during the course itself. Each article concludes with one or more examples illustrating application of the concepts covered to cardiovascular medicine and biology.

The conceptual basis of mathematics in cardiology IV: statistics and model fitting.

PubMed

Bates, Jason H T; Sobel, Burton E

2003-06-01

This is the fourth in a series of four articles developed for the readers of Coronary Artery Disease. Without language ideas cannot be articulated. What may not be so immediately obvious is that they cannot be formulated either. One of the essential languages of cardiology is mathematics. Unfortunately, medical education does not emphasize, and in fact, often neglects empowering physicians to think mathematically. Reference to statistics, conditional probability, multicompartmental modeling, algebra, calculus and transforms is common but often without provision of genuine conceptual understanding. At the University of Vermont College of Medicine, Professor Bates developed a course designed to address these deficiencies. The course covered mathematical principles pertinent to clinical cardiovascular and pulmonary medicine and research. It focused on fundamental concepts to facilitate formulation and grasp of ideas. This series of four articles was developed to make the material available for a wider audience. The articles will be published sequentially in Coronary Artery Disease. Beginning with fundamental axioms and basic algebraic manipulations they address algebra, function and graph theory, real and complex numbers, calculus and differential equations, mathematical modeling, linear system theory and integral transforms and statistical theory. The principles and concepts they address provide the foundation needed for in-depth study of any of these topics. Perhaps of even more importance, they should empower cardiologists and cardiovascular researchers to utilize the language of mathematics in assessing the phenomena of immediate pertinence to diagnosis, pathophysiology and therapeutics. The presentations are interposed with queries (by Coronary Artery Disease abbreviated as CAD) simulating the nature of interactions that occurred during the course itself. Each article concludes with one or more examples illustrating application of the concepts covered to cardiovascular medicine and biology.

The conceptual basis of mathematics in cardiology: (I) algebra, functions and graphs.

PubMed

Bates, Jason H T; Sobel, Burton E

2003-02-01

This is the first in a series of four articles developed for the readers of. Without language ideas cannot be articulated. What may not be so immediately obvious is that they cannot be formulated either. One of the essential languages of cardiology is mathematics. Unfortunately, medical education does not emphasize, and in fact, often neglects empowering physicians to think mathematically. Reference to statistics, conditional probability, multicompartmental modeling, algebra, calculus and transforms is common but often without provision of genuine conceptual understanding. At the University of Vermont College of Medicine, Professor Bates developed a course designed to address these deficiencies. The course covered mathematical principles pertinent to clinical cardiovascular and pulmonary medicine and research. It focused on fundamental concepts to facilitate formulation and grasp of ideas. This series of four articles was developed to make the material available for a wider audience. The articles will be published sequentially in Coronary Artery Disease. Beginning with fundamental axioms and basic algebraic manipulations they address algebra, function and graph theory, real and complex numbers, calculus and differential equations, mathematical modeling, linear system theory and integral transforms and statistical theory. The principles and concepts they address provide the foundation needed for in-depth study of any of these topics. Perhaps of even more importance, they should empower cardiologists and cardiovascular researchers to utilize the language of mathematics in assessing the phenomena of immediate pertinence to diagnosis, pathophysiology and therapeutics. The presentations are interposed with queries (by Coronary Artery Disease, abbreviated as CAD) simulating the nature of interactions that occurred during the course itself. Each article concludes with one or more examples illustrating application of the concepts covered to cardiovascular medicine and biology.

Galois groups of Schubert problems via homotopy computation

NASA Astrophysics Data System (ADS)

Leykin, Anton; Sottile, Frank

2009-09-01

Numerical homotopy continuation of solutions to polynomial equations is the foundation for numerical algebraic geometry, whose development has been driven by applications of mathematics. We use numerical homotopy continuation to investigate the problem in pure mathematics of determining Galois groups in the Schubert calculus. For example, we show by direct computation that the Galois group of the Schubert problem of 3-planes in mathbb{C}^8 meeting 15 fixed 5-planes non-trivially is the full symmetric group S_{6006} .

Introductory Life Science Mathematics and Quantitative Neuroscience Courses

PubMed Central

Olifer, Andrei

2010-01-01

We describe two sets of courses designed to enhance the mathematical, statistical, and computational training of life science undergraduates at Emory College. The first course is an introductory sequence in differential and integral calculus, modeling with differential equations, probability, and inferential statistics. The second is an upper-division course in computational neuroscience. We provide a description of each course, detailed syllabi, examples of content, and a brief discussion of the main issues encountered in developing and offering the courses. PMID:20810971

Do screencasts help to revise prerequisite mathematics? An investigation of student performance and perception

NASA Astrophysics Data System (ADS)

Loch, Birgit; Jordan, Camilla R.; Lowe, Tim W.; Mestel, Ben D.

2014-02-01

Basic calculus skills that are prerequisites for advanced mathematical studies continue to be a problem for a significant proportion of higher education students. While there are many types of revision material that could be offered to students, in this paper we investigate whether short, narrated video recordings of mathematical explanations (screencasts) are a useful tool to enhance student learning when revisiting prerequisite topics. We report on the outcomes of a study that was designed to both measure change in student performance before and after watching screencasts, and to capture students' perception of the usefulness of screencasts in their learning. Volunteers were recruited from students enrolled on an entry module for the Mathematics Master of Science programme at the Open University to watch two screencasts sandwiched between two online calculus quizzes. A statistical analysis of student responses to the quizzes shows that screencasts can have a positive effect on student performance. Further analysis of student feedback shows that student confidence was increased by watching the screencasts. Student views on the value of screencasts for their learning indicated that they appreciated being able to watch a problem being solved and explained by an experienced mathematician; hear the motivation for a particular problem-solving approach; engage more readily with the material being presented, thereby retaining it more easily. The positive student views and impact on student scores indicate that short screencasts could play a useful role in revising prerequisite mathematics.

Disinfection of Common Waterborne Pathogens

ERIC Educational Resources Information Center

Swim, Edward W.

2010-01-01

As part of an integrative learning experience at the end of a sophomore Calculus II course at the United States Military Academy, this project served as a multidisciplinary problem-solving exercise that explored the connections among mathematics, biology, and other fields of study. During a seven-lesson block of instruction, this module was…

The Effect of Math Modeling on Student's Emerging Understanding

ERIC Educational Resources Information Center

Sokolowski, Andrzej

2015-01-01

This study investigated the effects of applying mathematical modeling on revising students' preconception of the process of optimizing area enclosed by a string of a fixed length. A group of 28 high school pre-calculus students were immersed in modeling activity that included direct measurements, data collecting, and formulating algebraic…

Mathematics Placement at the University of Illinois

ERIC Educational Resources Information Center

Ahlgren Reddy, Alison; Harper, Marc

2013-01-01

Data from the ALEKS-based placement program at the University of Illinois is presented visually in several ways. The placement exam (an ALEKS assessment) contains precise item-specific information and the data show many interesting properties of the student populations of the placement courses, which include Precalculus, Calculus, and Business…

The Dreaded "Work" Problems Revisited: Connections through Problem Solving from Basic Fractions to Calculus

ERIC Educational Resources Information Center

Shore, Felice S.; Pascal, Matthew

2008-01-01

This article describes several distinct approaches taken by preservice elementary teachers to solving a classic rate problem. Their approaches incorporate a variety of mathematical concepts, ranging from proportions to infinite series, and illustrate the power of all five NCTM Process Standards. (Contains 8 figures.)

A Simple Interactive Software Package for Plotting, Animating, and Calculating

ERIC Educational Resources Information Center

Engelhardt, Larry

2012-01-01

We introduce a new open source (free) software package that provides a simple, highly interactive interface for carrying out certain mathematical tasks that are commonly encountered in physics. These tasks include plotting and animating functions, solving systems of coupled algebraic equations, and basic calculus (differentiating and integrating…

Fermat's Technique of Finding Areas under Curves

ERIC Educational Resources Information Center

Staples, Ed

2004-01-01

Perhaps next time teachers head towards the fundamental theorem of calculus in their classroom, they may wish to consider Fermat's technique of finding expressions for areas under curves, beautifully outlined in Boyer's History of Mathematics. Pierre de Fermat (1601-1665) developed some important results in the journey toward the discovery of the…

Module for Learning Integral Calculus with Maple: Lecturers' Views

ERIC Educational Resources Information Center

Awang, Tuan Salwani; Zakaria, Effandi

2012-01-01

Engineering technology students can attain a meaningful mathematics learning if they are allowed to actively participate in hands-on activities. However, the current dissemination of knowledge in the classroom still focuses on teacher-centered paradigm of teaching. A study to explore lecturers' views regarding a newly developed integral calculus…

Razalas' Grouping Method and Mathematics Achievement

ERIC Educational Resources Information Center

Salazar, Douglas A.

2015-01-01

This study aimed to raise the achievement level of students in Integral Calculus using Direct Instruction with Razalas' Method of Grouping. The study employed qualitative and quantitative analysis relative to data generated by the Achievement Test and Math journal with follow-up interview. Within the framework of the limitations of the study, the…

Adding It All Up: Reconceiving the Introduction of the Integral

ERIC Educational Resources Information Center

Jones, Steven R.

2013-01-01

Calculus instruction is an important topic for high school and college teachers alike. A prime target for attention is integration, which, unfortunately, students too often treat as a rote procedure. Understanding the integral better will support students' application of their mathematical knowledge to science, technology, and engineering…

A Model for Math Modeling

ERIC Educational Resources Information Center

Lin, Tony; Erfan, Sasan

2016-01-01

Mathematical modeling is an open-ended research subject where no definite answers exist for any problem. Math modeling enables thinking outside the box to connect different fields of studies together including statistics, algebra, calculus, matrices, programming and scientific writing. As an integral part of society, it is the foundation for many…

Revealing Educationally Critical Aspects of Rate

ERIC Educational Resources Information Center

Herbert, Sandra; Pierce, Robyn

2012-01-01

Rate (of change) is an important but complicated mathematical concept describing a ratio comparing two different numeric, measurable quantities. Research referring to students' difficulties with this concept spans more than 20 years. It suggests that problems experienced by some calculus students are likely a result of pre-existing limited or…

Who Takes College Algebra?

ERIC Educational Resources Information Center

Herriott, Scott R.; Dunbar, Steven R.

2009-01-01

The common understanding within the mathematics community is that the role of the college algebra course is to prepare students for calculus. Though exceptions are emerging, the curriculum of most college algebra courses and the content of most textbooks on the market both reflect that assumption. This article calls that assumption into question…

Stabilizing a Bicycle: A Modeling Project

ERIC Educational Resources Information Center

Pennings, Timothy J.; Williams, Blair R.

2010-01-01

This article is a project that takes students through the process of forming a mathematical model of bicycle dynamics. Beginning with basic ideas from Newtonian mechanics (forces and torques), students use techniques from calculus and differential equations to develop the equations of rotational motion for a bicycle-rider system as it tips from…

Testing Understanding and Understanding Testing.

ERIC Educational Resources Information Center

Pedersen, Jean; Ross, Peter

1985-01-01

Provides examples in which graphs are used in the statements of problems or in their solutions as a means of testing understanding of mathematical concepts. Examples (appropriate for a beginning course in calculus and analytic geometry) include slopes of lines and curves, quadratic formula, properties of the definite integral, and others. (JN)

Math 3310--Technical Mathematics I. Course Outline.

ERIC Educational Resources Information Center

New York Inst. of Tech., Old Westbury.

This document contains the course syllabus and 12 independent practice modules for a college pre-calculus designed as the first course in a two-semester sequence for students in a Bachelor of Technology program. The course emphasizes engineering technology applications and verbal problems. Topics include a review of elementary algebra; factoring…

Hermann-Bernoulli-Laplace-Hamilton-Runge-Lenz Vector.

ERIC Educational Resources Information Center

Subramanian, P. R.; And Others

1991-01-01

A way for students to refresh and use their knowledge in both mathematics and physics is presented. By the study of the properties of the "Runge-Lenz" vector the subjects of algebra, analytical geometry, calculus, classical mechanics, differential equations, matrices, quantum mechanics, trigonometry, and vector analysis can be reviewed. (KR)

The Geometric Mean Value Theorem

ERIC Educational Resources Information Center

de Camargo, André Pierro

2018-01-01

In a previous article published in the "American Mathematical Monthly," Tucker ("Amer Math Monthly." 1997; 104(3): 231-240) made severe criticism on the Mean Value Theorem and, unfortunately, the majority of calculus textbooks also do not help to improve its reputation. The standard argument for proving it seems to be applying…

Gains and Pitfalls of Quantifier Elimination as a Teaching Tool

ERIC Educational Resources Information Center

Oldenburg, Reinhard

2015-01-01

Quantifier Elimination is a procedure that allows simplification of logical formulas that contain quantifiers. Many mathematical concepts are defined in terms of quantifiers and especially in calculus their use has been identified as an obstacle in the learning process. The automatic deduction provided by quantifier elimination thus allows…

Enhancing students’ critical thinking skills through critical thinking assessment in calculus course

NASA Astrophysics Data System (ADS)

Zulfaneti; Edriati, S.; Mukhni

2018-01-01

This study aims to determine the development of students’ critical thinking skills through the implementation of critical thinking instruments in Calculus lectures. The instruments consist of observation sheets, critical thinking test, self-assessment, peer assessment and portfolio. The research was a qualitative research; with the participants were 53 first-year students who take Integral Calculus in Mathematics Education Department STKIP PGRI Sumatera Barat representing high-ability students, medium and low. The data in this study were collected by tests, interviews, observations and field notes. Data were analyzed descriptively; data reduction, data presentation, and conclusions. For testing the validity of data, it was used credibility test data by increasing persistence and triangulation. The results showed that in high-level students there is a change of ability from Critical enough to be Very Critical, in the students with moderate and low ability there is a change of ability from Uncritical to Critical. So it can be concluded that the assessment instruments have a good contribution and can improve the ability of critical thinking.

Towards a physics on fractals: Differential vector calculus in three-dimensional continuum with fractal metric

NASA Astrophysics Data System (ADS)

Balankin, Alexander S.; Bory-Reyes, Juan; Shapiro, Michael

2016-02-01

One way to deal with physical problems on nowhere differentiable fractals is the mapping of these problems into the corresponding problems for continuum with a proper fractal metric. On this way different definitions of the fractal metric were suggested to account for the essential fractal features. In this work we develop the metric differential vector calculus in a three-dimensional continuum with a non-Euclidean metric. The metric differential forms and Laplacian are introduced, fundamental identities for metric differential operators are established and integral theorems are proved by employing the metric version of the quaternionic analysis for the Moisil-Teodoresco operator, which has been introduced and partially developed in this paper. The relations between the metric and conventional operators are revealed. It should be emphasized that the metric vector calculus developed in this work provides a comprehensive mathematical formalism for the continuum with any suitable definition of fractal metric. This offers a novel tool to study physics on fractals.

Students Targeting Engineering and Physical Science (STEPS) at California State University Northridge (CSUN):Activities and Outcomes 2011-2016

NASA Astrophysics Data System (ADS)

Cadavid, A. C.; Pedone, V. A.; Horn, W.; Rich, H.

2016-12-01

The specific goal of STEPS at CSUN is to increase the number bachelor's degrees in STEM majors, particularly those in engineering, computer science, mathematics and the physical sciences. Prior to STEPS, only 33% of first-time freshmen in these majors graduated from CSUN within 6-7 years. We employ two main strategies: 1) fostering success in lower-division mathematics for freshmen and sophomores, 2) Summer Interdisciplinary Team Experience (SITE) for students transitioning to junior level courses. To improve success in mathematics, we have advanced initial placements in the foundational mathematics sequence by one or two semesters through improvements in the placement test (6-7% improvement) and have increased the first-time pass rate in foundational math courses through mandatory supplementary laboratories for at-risk students. Students who successfully complete the supplemental laboratories pass the lecture class at a higher rate than the total population of at-risk students (65% compared to 61%). Both approaches have been institutionalized. SITE targets students entering their junior years in a 3-week interdisciplinary team project that highlights problem solving and hands-on activities. Survey results of the 233 participants show that SITE: 1) maintained or increased desire to earn a STEM degree, 2) increased positive attitudes toward team-based problem solving, 3) increased understanding in how they will use their major in a career, and 4) increased interest in faculty-mentored research and industry internships. Our 5-year program is nearing completion and shows success in meeting our goal. We have measured a 9% point increase in the pass rate of Calculus I for post-STEPS cohorts compared to pre-STEPS cohorts. Failure to pass Calculus is a leading cause in non-completion of the majors targeted by STEPS. We have analyzed the graduation rates of two pre-STEPS cohorts that have had over 6 years to graduate. Both have a graduate rate of 28%. We expect that the 9% point increase in calculus passers will lead to a comparable increase in graduation rate, resulting in a 37% graduation rate for the post-STEPS cohorts.

Some applications of mathematics in theoretical physics - A review

DOE Office of Scientific and Technical Information (OSTI.GOV)

Bora, Kalpana

2016-06-21

Mathematics is a very beautiful subject−very much an indispensible tool for Physics, more so for Theoretical Physics (by which we mean here mainly Field Theory and High Energy Physics). These branches of Physics are based on Quantum Mechanics and Special Theory of Relativity, and many mathematical concepts are used in them. In this work, we shall elucidate upon only some of them, like−differential geometry, infinite series, Mellin transforms, Fourier and integral transforms, special functions, calculus, complex algebra, topology, group theory, Riemannian geometry, functional analysis, linear algebra, operator algebra, etc. We shall also present, some physics issues, where these mathematical toolsmore » are used. It is not wrong to say that Mathematics is such a powerful tool, without which, there can not be any Physics theory!! A brief review on our research work is also presented.« less

Some applications of mathematics in theoretical physics - A review

NASA Astrophysics Data System (ADS)

Bora, Kalpana

2016-06-01

Mathematics is a very beautiful subject-very much an indispensible tool for Physics, more so for Theoretical Physics (by which we mean here mainly Field Theory and High Energy Physics). These branches of Physics are based on Quantum Mechanics and Special Theory of Relativity, and many mathematical concepts are used in them. In this work, we shall elucidate upon only some of them, like-differential geometry, infinite series, Mellin transforms, Fourier and integral transforms, special functions, calculus, complex algebra, topology, group theory, Riemannian geometry, functional analysis, linear algebra, operator algebra, etc. We shall also present, some physics issues, where these mathematical tools are used. It is not wrong to say that Mathematics is such a powerful tool, without which, there can not be any Physics theory!! A brief review on our research work is also presented.

Problems with DNA

ERIC Educational Resources Information Center

Erickson, Keith A.; Franciszkowicz, Marc J.

2010-01-01

A modified version of this project was used during the final seven days of a year-long calculus sequence at the United States Military Academy to introduce students to the nature of integrative learning. Students from different majors were brought together in groups and spent the first few days going over the mathematics material presented here.…

Introduction to the Difference Calculus through the Fibonacci Numbers

ERIC Educational Resources Information Center

Shannon, A. G.; Atanassov, K. T.

2002-01-01

This note explores ways in which the Fibonacci numbers can be used to introduce difference equations as a prelude to differential equations. The rationale is that the formal aspects of discrete mathematics can provide a concrete introduction to the mechanisms of solving difference and differential equations without the distractions of the analytic…

The Instructional Network: Using Facebook to Enhance Undergraduate Mathematics Instruction

ERIC Educational Resources Information Center

Gregory, Peter; Gregory, Karen; Eddy, Erik

2014-01-01

Facebook is a website with over one billion users worldwide that is synonymous with social-networking. However, in this study, Facebook is used as an "instructional network". Two sections of an undergraduate calculus course were used to study the effects of participating in a Facebook group devoted solely to instruction. One section was…

The Art and Craft of Science

ERIC Educational Resources Information Center

Root-Bernstein, Robert; Root-Bernstein, Michele

2013-01-01

Walter Alvarez, a doctor and physiologist of some renown, decided to send his scientifically talented son, Luis, to an arts and crafts school where Luis took industrial drawing and woodworking instead of calculus. Luis Alvarez won the Nobel Prize in physics in 1968. Einstein was certainly not a standout in his mathematics and physics classes. Yet…

Student Connections between Algebraic and Graphical Polynomial Representations in the Context of a Polynomial Relation

ERIC Educational Resources Information Center

Adu-Gyamfi, Kwaku; Bossé, Michael J.; Chandler, Kayla

2017-01-01

When establishing connections among representations of associated mathematical concepts, students encounter different difficulties and successes along the way. The purpose of this study was to uncover information about and gain greater insight into how student processes connections. Pre-calculus students were observed and interviewed while…

Improving Student Learning in Calculus through Applications

ERIC Educational Resources Information Center

Young, C. Y.; Georgiopoulos, M.; Hagen, S. C.; Geiger, C. L.; Dagley-Falls, M. A.; Islas, A. L.; Ramsey, P. J.; Lancey, P. M.; Straney, R. A.; Forde, D. S.; Bradbury, E. E.

2011-01-01

Nationally only 40% of the incoming freshmen Science, Technology, Engineering and Mathematics (STEM) majors are successful in earning a STEM degree. The University of Central Florida (UCF) EXCEL programme is a National Science Foundation funded STEM Talent Expansion Programme whose goal is to increase the number of UCF STEM graduates. One of the…

Teaching Quantitative Management to Evening MBA Students.

ERIC Educational Resources Information Center

Libby, Barbara

1984-01-01

The author discusses the mathematics background of Masters of Business Administration (MBA) students and asks what math tools are necessary for an MBA. While she finds useful the ability to deal with linear and quadratic equations; interest, depreciation, and growth rates; and word problems, she concludes that calculus is of little use apart from…

Co-Calculus: Integrating the Academic and the Social

ERIC Educational Resources Information Center

Reinholz, Daniel L.

2017-01-01

Being part of a cohesive learning community supports retention and success in early mathematics courses. Yet, large, unwelcoming lectures stand in opposition to this goal, isolating students and pushing them away from STEM. This paper offers a comparative analysis of three efforts to build community amongst students, all situated within a single…

Assessing Clicker Examples versus Board Examples in Calculus

ERIC Educational Resources Information Center

Roth, Kimberly A.

2012-01-01

The combination of classroom voting system (clicker) questions and peer instruction has been shown to increase student learning. While implementations in large lectures have been around for a while, mathematics has been increasingly using clickers in classes of a smaller size. In Fall 2008, I conducted an experiment to measure the effect of…

Undergraduate Research: Mathematical Modeling of Mortgages

ERIC Educational Resources Information Center

Choi, Youngna; Spero, Steven

2010-01-01

In this article, we study financing in the real estate market and show how various types of mortgages can be modeled and analyzed. With only an introductory level of interest theory, finance, and calculus, we model and analyze three types of popular mortgages with real life examples that explain the background and inevitable outcome of the current…

Visualizing Three-Dimensional Calculus Concepts: The Study of a Manipulative's Effectiveness

ERIC Educational Resources Information Center

McGee, Daniel, Jr.; Moore-Russo, Deborah; Ebersole, Dennis; Lomen, David O.; Quintero, Maider Marin

2012-01-01

With the help of the National Science Foundation, the Department of Mathematics at the University of Puerto Rico in Mayaguez has developed a set of manipulatives to help students of science and engineering visualize concepts relating to points, surfaces, curves, contours, and vectors in three dimensions. This article will present the manipulatives…

Mathingo: Reviewing Calculus with Bingo Games

ERIC Educational Resources Information Center

Forman, Sean; Forman, Sylvia

2008-01-01

Games are a useful and fun way to review material in class, and encourage students to actively participate in review sessions. In fact, researchers have concluded that playing games can be an effective learning tool, especially in mathematics, in situations where the goal is to reinforce specific ideas or concepts. Two places this type of…

Examining Students' Generalizations of the Tangent Concept: A Theoretical Perspective

ERIC Educational Resources Information Center

Çekmez, Erdem; Baki, Adnan

2016-01-01

The concept of a tangent is important in understanding many topics in mathematics and science. Earlier studies on students' understanding of the concept of a tangent have reported that they have various misunderstandings and experience difficulties in transferring their knowledge about the tangent line from Euclidean geometry into calculus. In…

Impact of Context and Representation on Year 10 Students' Expression of Conceptions of Rate

ERIC Educational Resources Information Center

Herbert, Sandra

2010-01-01

Rate is an important, but difficult mathematical concept. More than twenty years of research, especially with calculus students, report difficulties with this concept. This paper reports on an alternative analysis, from the perspective of multiple representations and context, of interviews probing twenty Victorian Year 10 students' conceptions of…

The Minus Sign in Faraday's Law Revisited

ERIC Educational Resources Information Center

O'Sullivan, Colm; Hurley, Donal

2013-01-01

By introducing the mathematical concept of orientation, the significance of the minus sign in Faraday's law may be made clear to students with some knowledge of vector calculus. For many students, however, the traditional approach of treating the law as a relationship between positive scalars and of relying on Lenz's law to provide the information…

Senior Year Inviting More Math Choices

ERIC Educational Resources Information Center

Cavanagh, Sean

2008-01-01

When students at Prescott High School sign up to take math as seniors, not all of them will be wading into precalculus or calculus, with in-depth explorations of derivatives and trigonometric functions. Some will instead end up using mathematics to study the Electoral College, or the security of Internet passwords, or how delivery companies ship…

From "Work-and-Walk-By" to "Sherpa-at-Work"

ERIC Educational Resources Information Center

Drijvers, Paul

2011-01-01

Nowadays, many technological means are available to support teaching, such as the interactive whiteboard, class sets of laptop or netbook computers, and high speed internet access. For mathematics education there are advanced software packages for geometry, algebra, calculus, and statistics, which in many cases are available on line at no cost.…

Higher Integrability for Minimizers of the Mumford-Shah Functional

NASA Astrophysics Data System (ADS)

De Philippis, Guido; Figalli, Alessio

2014-08-01

We prove higher integrability for the gradient of local minimizers of the Mumford-Shah energy functional, providing a positive answer to a conjecture of De Giorgi (Free discontinuity problems in calculus of variations. Frontiers in pure and applied mathematics, North-Holland, Amsterdam, pp 55-62, 1991).

Personality Traits of Mathematically Advanced College Students.

ERIC Educational Resources Information Center

Spray, Kristina J.

This study examined how students with a minimum background of Calculus IV (n=17) differ from other college students (n=17) on personality traits as determined by the 16 PF, fifth edition. Significant differences were found on Factor A (Warmth), Factor B (Reasoning), Factor F (Liveliness) and Factor H (Social Boldness). Gender differences were also…

Calculus of Elementary Functions, Part II. Teacher's Commentary. Revised Edition.

ERIC Educational Resources Information Center

Herriot, Sarah T.; And Others

This course is intended for students who have a thorough knowledge of college preparatory mathematics, including algebra, axiomatic geometry, trigonometry, and analytic geometry. This teacher's guide is for Part II of the course. It is designed to follow Part I of the text. The guide contains background information, suggested instructional…

Calculus of Elementary Functions, Part I. Student Text. Revised Edition.

ERIC Educational Resources Information Center

Herriot, Sarah T.; And Others

This course is intended for students who have a thorough knowledge of college preparatory mathematics, including algebra, axiomatic geometry, trigonometry, and analytic geometry. This text, Part I, contains the first five chapters of the course and two appendices. Chapters included are: (1) Polynomial Functions; (2) The Derivative of a Polynomial…

Calculus of Elementary Functions, Part II. Student Text. Revised Edition.

ERIC Educational Resources Information Center

Herriot, Sarah T.; And Others

This course is intended for students who have a thorough knowledge of college preparatory mathematics, including algebra, axiomatic geometry, trigonometry, and analytic geometry. This text, Part II, contains material designed to follow Part I. Chapters included in this text are: (6) Derivatives of Exponential and Related Functions; (7) Area and…

Analysis of Frequency of Tests and Varying Feedback Delays in College Mathematics Achievement

ERIC Educational Resources Information Center

Townsend, Neal R.; Wheatley, Grayson H.

1975-01-01

Sixteen beginning analytic geometry and calculus classes (442 students) followed eight testing regimes for one academic quarter. Three aptitude subgroups were identified in each class. Classes to which daily quizzes were given achieved significantly higher on a specially constructed test than those which had only a single midterm examination.…

Bioreactors in Everyday Life: Ethanol and the Maize Craze

ERIC Educational Resources Information Center

Bowman, Silas

2010-01-01

This project served as a capstone event for the United States Military Academy sophomore Calculus II course. This multi-disciplinary problem-solving exercise motivated the link between math and biology and many other fields of study. The seven-lesson block of instruction was developed to show students how mathematics play a role in every…

Effect of Written Presentation on Performance in Introductory Physics

ERIC Educational Resources Information Center

Stewart, John; Ballard, Shawn

2010-01-01

This study examined the written work of students in the introductory calculus-based electricity and magnetism course at the University of Arkansas. The students' solutions to hourly exams were divided into a small set of countable features organized into three major categories, mathematics, language, and graphics. Each category was further divided…

Designing for Enhanced Conceptual Understanding in an Online Physics Course

ERIC Educational Resources Information Center

Dunlap, Joanna C.; Furtak, Thomas E.; Tucker, Susan A.

2009-01-01

The calculus-based, introductory physics course is the port of entry for any student interested in pursuing a college degree in the sciences, mathematics, or engineering. There is increasing demand for online delivery options that make the course more widely available, especially those that use best practices in student engagement. However,…

Problem Solving through an Optimization Problem in Geometry

ERIC Educational Resources Information Center

Poon, Kin Keung; Wong, Hang-Chi

2011-01-01

This article adapts the problem-solving model developed by Polya to investigate and give an innovative approach to discuss and solve an optimization problem in geometry: the Regiomontanus Problem and its application to football. Various mathematical tools, such as calculus, inequality and the properties of circles, are used to explore and reflect…

Rugby and Mathematics: A Surprising Link among Geometry, the Conics, and Calculus.

ERIC Educational Resources Information Center

Jones, Troy; Jackson, Steven

2001-01-01

Describes a rugby problem designed to help students understand the maximum-minimum situation. Presents a series of explorations that locate an optimal place for kicking the ball to maximize the angle at the goalposts. Uses interactive geometry software to construct a model of the situation. Includes a sample student activity. (KHR)

Hydrostatic Pressure Project: Linked-Class Problem-Based Learning in Engineering

ERIC Educational Resources Information Center

Davis, Freddie J.; Lockwood-Cooke, Pamela; Hunt, Emily M.

2011-01-01

Over the last few years, WTAMU Mathematics, Engineering and Science faculty has used interdisciplinary projects as the basis for implementation of a linked-class approach to Problem-Based Learning (PBL). A project that has significant relevance to engineering statics, fluid mechanics, and calculus is the Hydrostatic Pressure Project. This project…

An asymptotical machine

NASA Astrophysics Data System (ADS)

Cristallini, Achille

2016-07-01

A new and intriguing machine may be obtained replacing the moving pulley of a gun tackle with a fixed point in the rope. Its most important feature is the asymptotic efficiency. Here we obtain a satisfactory description of this machine by means of vector calculus and elementary trigonometry. The mathematical model has been compared with experimental data and briefly discussed.

Slope, Rate of Change, and Steepness: Do Students Understand These Concepts?

ERIC Educational Resources Information Center

Teuscher, Dawn; Reys, Robert E.

2010-01-01

How do mathematics teachers introduce the concepts of slope, rate of change, and steepness in their classrooms? Do students understand these concepts as interchangeable or regard them as three different ideas? In this article, the authors report the results of a study of high school Advanced Placement (AP) Calculus students who displayed…

The Aesthetic Calculus: Sex Appeal, Circuitry, and Invisibility

ERIC Educational Resources Information Center

Arntfield, Mike

2007-01-01

Since antiquity, ideas regarding true beauty have been usurped by the purview of mathematics. From the aesthetic "logic" of Aristotle to the instrumentalized brutality of the Final Solution and its methodical anthropometric measurements, we see how the symmetry of numbers has been used to quantify the bodily politic according to an empirical…

Pseudo Phase Plane and Fractional Calculus modeling of western global economic downturn

NASA Astrophysics Data System (ADS)

Tenreiro Machado, J. A.; Mata, Maria Eugénia

2015-05-01

This paper applies Pseudo Phase Plane (PPP) and Fractional Calculus (FC) mathematical tools for modeling world economies. A challenging global rivalry among the largest international economies began in the early 1970s, when the post-war prosperity declined. It went on, up to now. If some worrying threatens may exist actually in terms of possible ambitious military aggression, invasion, or hegemony, countries' PPP relative positions can tell something on the current global peaceful equilibrium. A global political downturn of the USA on global hegemony in favor of Asian partners is possible, but can still be not accomplished in the next decades. If the 1973 oil chock has represented the beginning of a long-run recession, the PPP analysis of the last four decades (1972-2012) does not conclude for other partners' global dominance (Russian, Brazil, Japan, and Germany) in reaching high degrees of similarity with the most developed world countries. The synergies of the proposed mathematical tools lead to a better understanding of the dynamics underlying world economies and point towards the estimation of future states based on the memory of each time series.

Deriving Laws from Ordering Relations

NASA Technical Reports Server (NTRS)

Knuth, Kevin H.

2004-01-01

The effect of Richard T. Cox's contribution to probability theory was to generalize Boolean implication among logical statements to degrees of implication, which are manipulated using rules derived from consistency with Boolean algebra. These rules are known as the sum rule, the product rule and Bayes Theorem, and the measure resulting from this generalization is probability. In this paper, I will describe how Cox s technique can be further generalized to include other algebras and hence other problems in science and mathematics. The result is a methodology that can be used to generalize an algebra to a calculus by relying on consistency with order theory to derive the laws of the calculus. My goals are to clear up the mysteries as to why the same basic structure found in probability theory appears in other contexts, to better understand the foundations of probability theory, and to extend these ideas to other areas by developing new mathematics and new physics. The relevance of this methodology will be demonstrated using examples from probability theory, number theory, geometry, information theory, and quantum mechanics.

A Study of Visualization for Mathematics Education

NASA Technical Reports Server (NTRS)

Daugherty, Sarah C.

2008-01-01

Graphical representations such as figures, illustrations, and diagrams play a critical role in mathematics and they are equally important in mathematics education. However, graphical representations in mathematics textbooks are static, Le. they are used to illustrate only a specific example or a limited set. of examples. By using computer software to visualize mathematical principles, virtually there is no limit to the number of specific cases and examples that can be demonstrated. However, we have not seen widespread adoption of visualization software in mathematics education. There are currently a number of software packages that provide visualization of mathematics for research and also software packages specifically developed for mathematics education. We conducted a survey of mathematics visualization software packages, summarized their features and user bases, and analyzed their limitations. In this survey, we focused on evaluating the software packages for their use with mathematical subjects adopted by institutions of secondary education in the United States (middle schools and high schools), including algebra, geometry, trigonometry, and calculus. We found that cost, complexity, and lack of flexibility are the major factors that hinder the widespread use of mathematics visualization software in education.

Global differential geometry: An introduction for control engineers

NASA Technical Reports Server (NTRS)

Doolin, B. F.; Martin, C. F.

1982-01-01

The basic concepts and terminology of modern global differential geometry are discussed as an introduction to the Lie theory of differential equations and to the role of Grassmannians in control systems analysis. To reach these topics, the fundamental notions of manifolds, tangent spaces, vector fields, and Lie algebras are discussed and exemplified. An appendix reviews such concepts needed for vector calculus as open and closed sets, compactness, continuity, and derivative. Although the content is mathematical, this is not a mathematical treatise but rather a text for engineers to understand geometric and nonlinear control.

If Gravity is Geometry, is Dark Energy just Arithmetic?

NASA Astrophysics Data System (ADS)

Czachor, Marek

2017-04-01

Arithmetic operations (addition, subtraction, multiplication, division), as well as the calculus they imply, are non-unique. The examples of four-dimensional spaces, R+4 and (- L/2, L/2)4, are considered where different types of arithmetic and calculus coexist simultaneously. In all the examples there exists a non-Diophantine arithmetic that makes the space globally Minkowskian, and thus the laws of physics are formulated in terms of the corresponding calculus. However, when one switches to the `natural' Diophantine arithmetic and calculus, the Minkowskian character of the space is lost and what one effectively obtains is a Lorentzian manifold. I discuss in more detail the problem of electromagnetic fields produced by a pointlike charge. The solution has the standard form when expressed in terms of the non-Diophantine formalism. When the `natural' formalsm is used, the same solution looks as if the fields were created by a charge located in an expanding universe, with nontrivially accelerating expansion. The effect is clearly visible also in solutions of the Friedman equation with vanishing cosmological constant. All of this suggests that phenomena attributed to dark energy may be a manifestation of a miss-match between the arithmetic employed in mathematical modeling, and the one occurring at the level of natural laws. Arithmetic is as physical as geometry.

Deducing the form factors for shear used in the calculus of the displacements based on strain energy methods. Mathematical approach for currently used shapes

NASA Astrophysics Data System (ADS)

Constantinescu, E.; Oanta, E.; Panait, C.

2017-08-01

The paper presents an initial study concerning the form factors for shear, for a rectangular and for a circular cross section, being used an analytical method and a numerical study. The numerical study considers a division of the cross section in small areas and uses the power of the definitions in order to compute the according integrals. The accurate values of the form factors are increasing the accuracy of the displacements computed by the use of the strain energy methods. The knowledge resulted from this study will be used for several directions of development: calculus of the form factors for a ring-type cross section of variable ratio of the inner and outer diameters, calculus of the geometrical characteristics of an inclined circular segment and, using a Bool algebra that operates with geometrical shapes, for an inclined circular ring segment. These shapes may be used to analytically define the geometrical model of a complex composite section, i.e. a ship hull cross section. The according calculus relations are also useful for the development of customized design commands in CAD commercial applications. The paper is a result of the long run development of original computer based instruments in engineering of the authors.

A delivery mode study: The effect of self-paced video learning on first-year college students' achievement in calculus

NASA Astrophysics Data System (ADS)

Oktaviyanthi, Rina; Herman, Tatang

2016-10-01

In this paper, the effect of two different modes of deliver are proposed. The use of self-paced video learning and conventional learning methods in mathematics are compared. The research design classified as a quasi-experiment. The participants were 80 students in the first-year college and divided into two groups. One group as an experiment class received self-paced video learning method and the other group as a control group taught by conventional learning method. Pre and posttest were employed to measure the students' achievement, while questionnaire and interviews were applied to support the pre and posttest data. Statistical analysis included the independent samples t-test showed differences (p < 0.05) in posttest between the experimental and control groups, it means that the use of self-paced video contributed on students' achievement and students' attitudes. In addition, related to corresponding to the students' answer, there are five positive gains in using self-paced video in learning Calculus, such as appropriate learning for both audio and visual of students' characteristics, useful to learn Calculus, assisting students to be more engaging and paying attention in learning, helping students in making the concepts of Calculus are visible, interesting media and motivating students to learn independently.

Differential forms for scientists and engineers

NASA Astrophysics Data System (ADS)

Blair Perot, J.; Zusi, Christopher J.

2014-01-01

This paper is a review of a number of mathematical concepts from differential geometry and exterior calculus that are finding increasing application in the numerical solution of partial differential equations. The objective of the paper is to introduce the scientist/ engineer to some of these ideas via a number of concrete examples in 2, 3, and 4 dimensions. The goal is not to explain these ideas with mathematical precision but to present concrete examples and enable a physical intuition of these concepts for those who are not mathematicians. The objective of this paper is to provide enough context so that scientist/engineers can interpret, implement, and understand other works which use these elegant mathematical concepts.

Bridging a cultural gap

NASA Astrophysics Data System (ADS)

Leviatan, Talma

2008-09-01

There has been a broad wave of change in tertiary calculus courses in the past decade. However, the much-needed change in tertiary pre-calculus programmes—aimed at bridging the gap between high-school mathematics and tertiary mathematics—is happening at a far slower pace. Following a discussion on the nature of the gap and the objectives of a potential bridging programme, this paper aims at demonstrating that the gap can be bridged, by presenting an ongoing modular bridging programme especially designed for the diverse types of student populations in teachers training colleges. We also present here some innovative teaching and assessment methods that were judged essential for the success of these programmes—focusing mainly on the "Questionnaire Based Instruction Method". Finally we suggest directions of follow up and research.

Authority in an Agency-Centered, Inquiry-Based University Calculus Classroom

ERIC Educational Resources Information Center

Gerson, Hope; Bateman, Elizabeth

2010-01-01

Authority roles among teachers and students have traditionally been hierarchal and centered with the expertise and power of the teacher limiting opportunities for students to act with autonomy to build and justify mathematics. In this paper we discuss authority roles for teachers and students that have been realized in an inquiry-based university,…

An Extension of the Mean Value Theorem for Integrals

ERIC Educational Resources Information Center

Khalili, Parviz; Vasiliu, Daniel

2010-01-01

In this note we present an extension of the mean value theorem for integrals. The extension we consider is motivated by an older result (here referred as Corollary 2), which is quite classical for the literature of Mathematical Analysis or Calculus. We also show an interesting application for computing the sum of a harmonic series.

Fractional Diffusion Equations and Anomalous Diffusion

NASA Astrophysics Data System (ADS)

Evangelista, Luiz Roberto; Kaminski Lenzi, Ervin

2018-01-01

Preface; 1. Mathematical preliminaries; 2. A survey of the fractional calculus; 3. From normal to anomalous diffusion; 4. Fractional diffusion equations: elementary applications; 5. Fractional diffusion equations: surface effects; 6. Fractional nonlinear diffusion equation; 7. Anomalous diffusion: anisotropic case; 8. Fractional Schrödinger equations; 9. Anomalous diffusion and impedance spectroscopy; 10. The Poisson–Nernst–Planck anomalous (PNPA) models; References; Index.

Predicting Flu Season Requirements: An Undergraduate Modeling Project

ERIC Educational Resources Information Center

Kramlich, Gary R., II; Braunstein Fierson, Janet L.; Wright, J. Adam

2010-01-01

This project was designed to be used in a freshman calculus class whose students had already been introduced to logistic functions and basic data modeling techniques. It need not be limited to such an audience, however; it has also been implemented in a topics in mathematics class for college upperclassmen. Originally intended to be presented in…

There's More to the Multimedia Effect than Meets the Eye: Is Seeing Pictures Believing?

ERIC Educational Resources Information Center

Ögren, Magnus; Nyström, Marcus; Jarodzka, Halszka

2017-01-01

Textbooks in applied mathematics often use graphs to explain the meaning of formulae, even though their benefit is still not fully explored. To test processes underlying this assumed multimedia effect we collected performance scores, eye movements, and think-aloud protocols from students solving problems in vector calculus with and without graphs.…

Calculus Students' Representation Use in Group-Work and Individual Settings

ERIC Educational Resources Information Center

Zazkis, Dov

2013-01-01

The study of student representation use and specifically the distinction between analytic and visual representations has fueled a long line of mathematics education literature that began more than 35 years ago. This literature can be partitioned into two bodies of work, one that is primarily cognitive and one that is primarily social. In spite of…

Climate Modeling in the Calculus and Differential Equations Classroom

ERIC Educational Resources Information Center

Kose, Emek; Kunze, Jennifer

2013-01-01

Students in college-level mathematics classes can build the differential equations of an energy balance model of the Earth's climate themselves, from a basic understanding of the background science. Here we use variable albedo and qualitative analysis to find stable and unstable equilibria of such a model, providing a problem or perhaps a…

Evaluating the Incorporation of Technology and Application Projects in the Higher Education Mathematics Classroom

ERIC Educational Resources Information Center

Schreyer-Bennethum, Lynn; Albright, Leonard

2011-01-01

We report the qualitative and quantitative results of incorporating interdisciplinary application projects and increasing the use of teaching with technology into Calculus I, II and III at the University of Colorado Denver. Minimal changes were made to the curriculum and minimal time was required of instructors to make the changes. Instructors…

Corvettes, Curve Fitting, and Calculus

ERIC Educational Resources Information Center

Murawska, Jaclyn M.; Nabb, Keith A.

2015-01-01

Sometimes the best mathematics problems come from the most unexpected situations. Last summer, a Corvette raced down a local quarter-mile drag strip. The driver, a family member, provided the spectators with time and distance-traveled data from his time slip and asked "Can you calculate how many seconds it took me to go from 0 to 60…

Learner-Content, Learner-Instructor, and Learner-Learner Interaction in a Web-Enhanced, Internet Videoconference AP Calculus Course

ERIC Educational Resources Information Center

Einfeld, Dana Hobbs

2014-01-01

The purpose of this action research was to investigate how the use of technology promotes interaction to foster high school students' mathematical understanding. This mixed method study is guided by social-constructivist theory (Vygotsky, 1978) and framed within Moore's (1989) model of learner-content, learner-instructor, and learner-learner…

Signifying the Accumulation Graph in a Dynamic and Multi-Representation Environment

ERIC Educational Resources Information Center

Yerushalmy, Michal; Swidan, Osama

2012-01-01

The present study focuses on the accumulation process involved in the integration of a single-variable function. Observing the work of two high-school calculus students who had not yet learned any other integral-related ideas, we analyze the emergence of the semiotic relationship between personal and mathematical meanings, as expressed through the…

The Harmonic Series Diverges Again and Again

ERIC Educational Resources Information Center

Kifowit, Steven J.; Stamps, Terra A.

2006-01-01

The harmonic series is one of the most celebrated infinite series of mathematics. A quick glance at a variety of modern calculus textbooks reveals that there are two very popular proofs of the divergence of the harmonic series. In this article, the authors survey these popular proofs along with many other proofs that are equally simple and…

Active Learning in a Math for Liberal Arts Classroom

ERIC Educational Resources Information Center

Lenz, Laurie

2015-01-01

Inquiry-based learning is a topic of growing interest in the mathematical community. Much of the focus has been on using these methods in calculus and higher-level classes. This article describes the design and implementation of a set of inquiry-based learning activities in a Math for Liberal Arts course at a small, private, Catholic college.…

Calculus, Part 3, Teacher's Commentary, Unit No. 71. Revised Edition.

ERIC Educational Resources Information Center

Beck, A.; And Others

This is part three of a three-part manual for teachers using SMSG high school text materials. The overall purpose for each of the chapters is described and the mathematical development detailed. Background information for key concepts and answers for all exercises in each chapter are provided. Chapter topics include: (1) vectors and curves; (2)…

Introducing Computational Thinking through Hands-on Projects Using R with Applications to Calculus, Probability and Data Analysis

ERIC Educational Resources Information Center

Benakli, Nadia; Kostadinov, Boyan; Satyanarayana, Ashwin; Singh, Satyanand

2017-01-01

The goal of this paper is to promote computational thinking among mathematics, engineering, science and technology students, through hands-on computer experiments. These activities have the potential to empower students to learn, create and invent with technology, and they engage computational thinking through simulations, visualizations and data…

Geometric Series: A New Solution to the Dog Problem

ERIC Educational Resources Information Center

Dion, Peter; Ho, Anthony

2013-01-01

This article describes what is often referred to as the dog, beetle, mice, ant, or turtle problem. Solutions to this problem exist, some being variations of each other, which involve mathematics of a wide range of complexity. Herein, the authors describe the intuitive solution and the calculus solution and then offer a completely new solution…

A Brief but Important Note on the Product Rule

ERIC Educational Resources Information Center

Merrotsy, Peter

2016-01-01

The leap into the wonderful world of differential calculus can be daunting for many students, and hence it is important to ensure that the landing is as gentle as possible. When the product rule, for example, is met in the "Australian Curriculum: Mathematics", sound pedagogy would suggest developing and presenting the result in a form…

Modeling a Day in the Life of a Diabetic

ERIC Educational Resources Information Center

Brod, Ryan; Gomber, John; Mendoza, Jurelle; Roginski, Jonathan; Smith, Tyler

2010-01-01

The material presented here was used for a semester-long capstone project for a first semester freshman course entitled Mathematical Modeling and Introduction to Calculus. The goals for the students in this work were twofold: first, enable the students to gain insight into an actual problem that affects millions of people in the United States and…

Deriving the Regression Equation without Using Calculus

ERIC Educational Resources Information Center

Gordon, Sheldon P.; Gordon, Florence S.

2004-01-01

Probably the one "new" mathematical topic that is most responsible for modernizing courses in college algebra and precalculus over the last few years is the idea of fitting a function to a set of data in the sense of a least squares fit. Whether it be simple linear regression or nonlinear regression, this topic opens the door to applying the…

A Decade of COMPASS: Improving High School Mathematics Education through a National Curriculum Implementation Center

ERIC Educational Resources Information Center

Allen, Kasi; St. John, Mark; Tambe, Pamela

2009-01-01

Back in 1992, the National Science Foundation (NSF) awarded grants to five curriculum development teams and charged them with the task of starting over. Five years later, each of the development teams had produced an innovative and "integrated" curriculum. All represented notable departures from the commonly encountered, calculus-driven high…

Pre-University Students' Errors in Integration of Rational Functions and Implications for Classroom Teaching

ERIC Educational Resources Information Center

Yee, Ng Kin; Lam, Toh Tin

2008-01-01

This paper reports on students' errors in performing integration of rational functions, a topic of calculus in the pre-university mathematics classrooms. Generally the errors could be classified as those due to the students' weak algebraic concepts and their lack of understanding of the concept of integration. With the students' inability to link…

An Uncommon Approach to a Common Algebraic Error

ERIC Educational Resources Information Center

Rossi, Paul S.

2008-01-01

The basic rules of elementary algebra can often appear beyond the grasp of many students. Even though most subjects, including calculus, prove to be more difficult, it is the simple rules of algebra that continue to be the "thorn in the side" of many mathematics students. In this paper we present a result intended to help students achieve a…

The Curl of a Vector Field: Beyond the Formula

ERIC Educational Resources Information Center

Burch, Kimberly Jordan; Choi, Youngna

2006-01-01

It has been widely acknowledged that there is some discrepancy in the teaching of vector calculus in mathematics courses and other applied fields. The curl of a vector field is one topic many students can calculate without understanding its significance. In this paper, we explain the origin of the curl after presenting the standard mathematical…

Enabling and Integrating Online Formative Assessment in a Flipped Calculus Course

ERIC Educational Resources Information Center

Schroeder, Larissa Bucchi; Dorn, Brian

2016-01-01

The increased use of videos in mathematics courses means that direct instruction, traditionally part of class lectures, now often occurs outside of the classroom. Although students come to class with some baseline level of understanding, instructors lose opportunities to assess students' understanding of content as it is seen for the first time.…

Techniques of Differentiation and Integration, Mathematics (Experimental): 5297.27.

ERIC Educational Resources Information Center

Forrester, Gary B.

This guidebook on minimum course content was designed for students who have mastered the skills and concepts of analytic geometry. It is a short course in the basic techniques of calculus recommended for the student who has need of these skills in other courses such as beginning physics, economics or statistics. The course does not intend to teach…

The Pendulum: A Paradigm for the Linear Oscillator

ERIC Educational Resources Information Center

Newburgh, Ronald

2004-01-01

The simple pendulum is a model for the linear oscillator. The usual mathematical treatment of the problem begins with a differential equation that one solves with the techniques of the differential calculus, a formal process that tends to obscure the physics. In this paper we begin with a kinematic description of the motion obtained by experiment…

Slow off the Mark: Elementary School Teachers and the Crisis in STEM Education

ERIC Educational Resources Information Center

Epstein, Diana; Miller, Raegen T.

2011-01-01

Prospective teachers can typically obtain a license to teach elementary school without taking a rigorous college-level STEM class such as calculus, statistics, or chemistry, and without demonstrating a solid grasp of mathematics knowledge, scientific knowledge, or the nature of scientific inquiry. This is not a recipe for ensuring students have…

Basic Math Skills and Performance in an Introductory Economics Class

ERIC Educational Resources Information Center

Ballard, Charles L.; Johnson, Marianne F.

2004-01-01

The authors measure math skills with a broader set of explanatory variables than have been used in previous studies. To identify what math skills are important for student success in introductory microeconomics, they examine (1) the student's score on the mathematics portion of the ACT Assessment Test, (2) whether the student has taken calculus,…

Technology Tips: Building Interactive Demonstrations with Sage

ERIC Educational Resources Information Center

Murray, Maura

2013-01-01

Sage is an open-source software package that can be used in many different areas of mathematics, ranging from algebra to calculus and beyond. One of the most exciting pedagogical features of Sage (http://www.sagemath.org) is its ability to create interacts--interactive examples that can be used in a classroom demonstration or by students in a…

Vectors, Change of Basis and Matrix Representation: Onto-Semiotic Approach in the Analysis of Creating Meaning

ERIC Educational Resources Information Center

Montiel, Mariana; Wilhelmi, Miguel R.; Vidakovic, Draga; Elstak, Iwan

2012-01-01

In a previous study, the onto-semiotic approach was employed to analyse the mathematical notion of different coordinate systems, as well as some situations and university students' actions related to these coordinate systems in the context of multivariate calculus. This study approaches different coordinate systems through the process of change of…

Douglas Butler Uses Autograph to Explore the Geometry of Calculus

ERIC Educational Resources Information Center

Butler, Douglas

2012-01-01

In short, this is a "master class". The learning and teaching of mathematics can be revolutionised with the creative use of dynamic software is an oft quoted mantra. Here, this mantra is exemplified through the documented experiences of using Autograph to enliven, to extend, and to foster the understanding of differentiation and integration. The…

Equivalent Vectors

ERIC Educational Resources Information Center

Levine, Robert

2004-01-01

The cross-product is a mathematical operation that is performed between two 3-dimensional vectors. The result is a vector that is orthogonal or perpendicular to both of them. Learning about this for the first time while taking Calculus-III, the class was taught that if AxB = AxC, it does not necessarily follow that B = C. This seemed baffling. The…

The Relationship between Gender and Students' Attitude and Experience of Using a Computer Algebra System

ERIC Educational Resources Information Center

Ocak, Mehmet

2008-01-01

This correlational study examined the relationship between gender and the students' attitude and prior knowledge of using one of the mathematical software programs (MATLAB). Participants were selected from one community college, one state university and one private college. Students were volunteers from three Calculus I classrooms (one class from…

Do Dogs Know Related Rates Rather than Optimization?

ERIC Educational Resources Information Center

Perruchet, Pierre; Gallego, Jorge

2006-01-01

Although dogs seemingly follow the optimal path where they get to a ball thrown into the water, they certainly do not know the minimization function proposed in the calculus books. Trading the optimization problem for a related rates problem leads to a mathematically identical solution, which, it is argued here, is a more plausible model for the…

A Report on the Present Status of Engineering Mathematics Test (EMaT)

NASA Astrophysics Data System (ADS)

Watanabe, Toshimasa; Takafuji, Daisuke

The aim of Engineering Mathematics Test (EMaT) is to make sure what essentials in curriculum of Engineering Mathematics is, and to assess university students’ core academic competence and achievement of Engineering Mathematics, helping assurance of students’ academic ability. It is useful for professors to evaluate teaching effect of the classes, and this evaluation would help them improve curricula. Scores can be available for both graduate school entrance examinations and employment tests, leading to selecting persons with basic academic ability in Engineering Mathematics. The scope includes fundamentals in Calculus, Linear Algebra, Differential Equations, and Probability and Statistics. It is open to all students free of charge, and is annually given once in December. In 2007, 2,396 students from 35 universities took EMaT, and the total number of students who have taken EMaT in these 5 years is 6,240.

Why do I need to know this? Optics/photonics problem-based learning in the math classroom

NASA Astrophysics Data System (ADS)

Donnelly, Matthew J.; Donnelly, Judith F.; Donnelly, Stephanie

2017-08-01

A common complaint of engineering managers is that new employees at all levels, technician through engineer, tend to have rote calculation ability but are unable to think critically and use structured problem solving techniques to apply mathematical concepts. Further, they often have poor written and oral communication skills and difficulty working in teams. Ironically, a common question of high school mathematics students is "Why do I need to know this?" In this paper we describe a project using optics/photonics and Problem Based Learning (PBL) to address these issues in a high school calculus classroom.

Stochastic differential equations

DOE Office of Scientific and Technical Information (OSTI.GOV)

Sobczyk, K.

1990-01-01

This book provides a unified treatment of both regular (or random) and Ito stochastic differential equations. It focuses on solution methods, including some developed only recently. Applications are discussed, in particular an insight is given into both the mathematical structure, and the most efficient solution methods (analytical as well as numerical). Starting from basic notions and results of the theory of stochastic processes and stochastic calculus (including Ito's stochastic integral), many principal mathematical problems and results related to stochastic differential equations are expounded here for the first time. Applications treated include those relating to road vehicles, earthquake excitations and offshoremore » structures.« less

Quantum-Like Models for Decision Making in Psychology and Cognitive Science

NASA Astrophysics Data System (ADS)

Khrennikov, Andrei.

2009-02-01

We show that (in contrast to rather common opinion) the domain of applications of the mathematical formalism of quantum mechanics is not restricted to physics. This formalism can be applied to the description of various quantum-like (QL) information processing. In particular, the calculus of quantum (and more general QL) probabilities can be used to explain some paradoxical statistical data which was collected in psychology and cognitive science. The main lesson of our study is that one should sharply distinguish the mathematical apparatus of QM from QM as a physical theory. The domain of application of the mathematical apparatus is essentially wider than quantum physics. Quantum-like representation algorithm, formula of total probability, interference of probabilities, psychology, cognition, decision making.

A computable expression of closure to efficient causation.

PubMed

Mossio, Matteo; Longo, Giuseppe; Stewart, John

2009-04-07

In this paper, we propose a mathematical expression of closure to efficient causation in terms of lambda-calculus; we argue that this opens up the perspective of developing principled computer simulations of systems closed to efficient causation in an appropriate programming language. An important implication of our formulation is that, by exhibiting an expression in lambda-calculus, which is a paradigmatic formalism for computability and programming, we show that there are no conceptual or principled problems in realizing a computer simulation or model of closure to efficient causation. We conclude with a brief discussion of the question whether closure to efficient causation captures all relevant properties of living systems. We suggest that it might not be the case, and that more complex definitions could indeed create crucial some obstacles to computability.

The Effect of a Constructivist Learning Environment on the Limit Concept among Mathematics Student Teachers

ERIC Educational Resources Information Center

Bukova-Guzel, Esra

2007-01-01

The purpose of this study is to design a constructivist learning environment that helps learning the limit concept. The study is a pretest-posttest quasi-experimental research. The control and the experimental groups were chosen from the students attending a calculus course. Worksheets were used to assess students' learning of the limit concept.…

CAS or Pen-and-Paper: Factors That Influence Students' Choices

ERIC Educational Resources Information Center

Cameron, Scott; Ball, Lynda

2015-01-01

This paper reports on a study of choices about the use of a computer algebra system (CAS) or pen-and-paper (p&p) by a class of seven Year 11 Mathematical Methods (CAS) students as they completed a calculus worksheet. Factors that influenced students' choices are highlighted by comparing and contrasting the use of CAS and p&p between…

Assessment of Factors Impacting Success for Incoming College Engineering Students in a Summer Bridge Program

ERIC Educational Resources Information Center

Reisel, John R.; Jablonski, Marissa; Hosseini, Hossein; Munson, Ethan

2012-01-01

A summer bridge program for incoming engineering and computer science freshmen has been used at the University of Wisconsin-Milwaukee from 2007 to 2010. The primary purpose of this program has been to improve the mathematics course placement for incoming students who initially place into a course below Calculus I on the math placement examination.…

How Can Students Generalize the Chain Rule? The Roles of Abduction in Mathematical Modeling

ERIC Educational Resources Information Center

Park, Jin Hyeong; Lee, Kyeong-Hwa

2016-01-01

The purpose of this study is to design a modeling task to facilitate students' inquiries into the chain rule in calculus and to analyze the results after implementation of the task. In this study, we take a modeling approach to the teaching and learning of the chain rule by facilitating the generalization of students' models and modeling…

Comparing the Impact of Traditional and Modeling College Algebra Courses on Student Performance in Survey of Calculus

ERIC Educational Resources Information Center

West, Jerry G.

2013-01-01

Students in higher education deserve opportunities to succeed and learning environments which maximize success. Mathematics courses can create a barrier for success for some students. College algebra is a course that serves as a gateway to required courses in many bachelor's degree programs. The content in college algebra should serve to…

University Students' Retention of Derivative Concepts 14 Months after the Course: Influence of "Met-Befores" and "Met-Afters"

ERIC Educational Resources Information Center

Jukic, Ljerka; Dahl, Bettina

2012-01-01

This article reports the concluding part of a larger study on retention of key procedural and conceptual concepts in differential and integral calculus among Croatian and Danish university students in non-mathematics study programmes. The first parts of the study examined the retention of the students' knowledge through a questionnaire testing…

Didactic Situations and Didactical Engineering in University Mathematics: Cases from the Study of Calculus and Proof

ERIC Educational Resources Information Center

González-Martín, Alejandro S.; Bloch, Isabelle; Durand-Guerrier, Viviane; Maschietto, Michela

2014-01-01

This paper discusses the use of the "Theory of Didactic Situations" (TDS) at university level, paying special attention to the constraints and specificities of its use at this level. We begin by presenting the origins and main tenets of this approach, and discuss how these tenets are used towards the design of "Didactical…

Solar Radiation and the UV Index: An Application of Numerical Integration, Trigonometric Functions, Online Education and the Modelling Process

ERIC Educational Resources Information Center

Downs, Nathan; Parisi, Alfio V.; Galligan, Linda; Turner, Joanna; Amar, Abdurazaq; King, Rachel; Ultra, Filipina; Butler, Harry

2016-01-01

A short series of practical classroom mathematics activities employing the use of a large and publicly accessible scientific data set are presented for use by students in years 9 and 10. The activities introduce and build understanding of integral calculus and trigonometric functions through the presentation of practical problem solving that…

A Quantitative Analysis of the Relationship between an Online Homework System and Student Achievement in Pre-Calculus

ERIC Educational Resources Information Center

Babaali, Parisa; Gonzalez, Lidia

2015-01-01

Supporting student success in entry-level mathematics courses at the undergraduate level has and continues to be a challenge. Recently we have seen an increased reliance on technological supports including software to supplement more traditional in-class instruction. In this paper, we explore the effects on student performance of the use of a…

Cognitive Development of Applying the Chain Rule through Three Worlds of Mathematics

ERIC Educational Resources Information Center

Kabael, Tangul Uygur

2010-01-01

The derivative of a composite function, taken with the chain rule is one of the important notions in calculus. This paper describes a study conducted in Turkey that shows that the chain rule was given with the formula in function notation and/or the Leibniz notation without relating these formulas to life-related problem situations in the…

Shapes of the Graphs of Fourth-Degree Polynomials in Terms of Their Coefficients

ERIC Educational Resources Information Center

Flesher, Tatyana; Holder, Eleanor

2007-01-01

One of the main problems in undergraduate research in pure mathematics is that of determining a problem that is, at once, interesting to and capable of solution by a student who has completed only the calculus sequence. It is also desirable that the problem should present something new, since novelty and originality greatly increase the enthusiasm…

Designing Learning Strategy to Improve Undergraduate Students' Problem Solving in Derivatives and Integrals: A Conceptual Framework

ERIC Educational Resources Information Center

Hashemi, Nourooz; Abu, Mohd Salleh; Kashefi, Hamidreza; Mokhtar, Mahani; Rahimi, Khadijeh

2015-01-01

Derivatives and integrals are two important concepts of calculus which are precondition topics for most of mathematics courses and other courses in different fields of studies. A majority of students at the undergraduate level have to master derivatives and integrals if they want to be successful in their studies However, students encounter…

Developing Essential Understanding of Functions for Teaching Mathematics in Grades 9-12

ERIC Educational Resources Information Center

Lloyd, Gwendolyn; Beckmann, Sybilla; Zbiek, Rose Mary; Cooney, Thomas

2010-01-01

Are sequences functions? What can't the popular "vertical line test" be applied in some cases to determine if a relation is a function? How does the idea of rate of change connect with simpler ideas about proportionality as well as more advanced topics in calculus? Helping high school students develop a robust understanding of functions requires…

Linking Computer Algebra Systems and Paper-and-Pencil Techniques To Support the Teaching of Mathematics.

ERIC Educational Resources Information Center

van Herwaarden, Onno A.; Gielen, Joseph L. W.

2002-01-01

Focuses on students showing a lack of conceptual insight while using computer algebra systems (CAS) in the setting of an elementary calculus and linear algebra course for first year university students in social sciences. The use of a computer algebra environment has been incorporated into a more traditional course but with special attention on…

Weekly Online Quizzes to a Mathematics Course for Engineering Students

ERIC Educational Resources Information Center

Gaspar Martins, Sandra

2017-01-01

A set of weekly optional online quizzes was used with 104 students on a Multivariable Calculus course (MC), via the Moodle online system. These quizzes contributed a maximum of two extra points, and this was awarded if the student scored more than 9 points (out of 20) on the exam. All the students got the same questions and could resubmit the…

Numerical methods for stochastic differential equations

NASA Astrophysics Data System (ADS)

Kloeden, Peter; Platen, Eckhard

1991-06-01

The numerical analysis of stochastic differential equations differs significantly from that of ordinary differential equations due to the peculiarities of stochastic calculus. This book provides an introduction to stochastic calculus and stochastic differential equations, both theory and applications. The main emphasise is placed on the numerical methods needed to solve such equations. It assumes an undergraduate background in mathematical methods typical of engineers and physicists, through many chapters begin with a descriptive summary which may be accessible to others who only require numerical recipes. To help the reader develop an intuitive understanding of the underlying mathematicals and hand-on numerical skills exercises and over 100 PC Exercises (PC-personal computer) are included. The stochastic Taylor expansion provides the key tool for the systematic derivation and investigation of discrete time numerical methods for stochastic differential equations. The book presents many new results on higher order methods for strong sample path approximations and for weak functional approximations, including implicit, predictor-corrector, extrapolation and variance-reduction methods. Besides serving as a basic text on such methods. the book offers the reader ready access to a large number of potential research problems in a field that is just beginning to expand rapidly and is widely applicable.

A fresh look at the catenary

NASA Astrophysics Data System (ADS)

Behroozi, F.

2014-09-01

A hanging chain takes the familiar form known as the catenary which is one of the most ubiquitous curves students encounter in their daily life. Yet most introductory physics and mathematics texts ignore the subject entirely. In more advanced texts the catenary equation is usually derived as an application of the calculus of variations. Although the variational approach is mathematically elegant, it is suitable for more advanced students. Here we derive the catenary equation in special and rectangular coordinates by considering the equilibrium conditions for an element of the hanging chain and without resorting to the calculus of variations. One advantage of this approach is its simplicity which makes it accessible to undergraduate students; another is the concurrent derivation of a companion equation which gives the tension along the chain. These solutions provide an excellent opportunity for undergraduates to explore the underlying physics. One interesting result is that the shape of a hanging chain does not depend on its linear mass density or on the strength of the gravitational field. Therefore, within a scale factor, all catenaries are copies of the same universal curve. We give the functional dependence of the scale factor on the length and terminal angle of the hanging chain.

Theoretical distribution of gutta-percha within root canals filled using cold lateral compaction based on numeric calculus.

PubMed

Min, Yi; Song, Ying; Gao, Yuan; Dummer, Paul M H

2016-08-01

This study aimed to present a new method based on numeric calculus to provide data on the theoretical volume ratio of voids when using the cold lateral compaction technique in canals with various diameters and tapers. Twenty-one simulated mathematical root canal models were created with different tapers and sizes of apical diameter, and were filled with defined sizes of standardized accessory gutta-percha cones. The areas of each master and accessory gutta-percha cone as well as the depth of their insertion into the canals were determined mathematically in Microsoft Excel. When the first accessory gutta-percha cone had been positioned, the residual area of void was measured. The areas of the residual voids were then measured repeatedly upon insertion of additional accessary cones until no more could be inserted in the canal. The volume ratio of voids was calculated through measurement of the volume of the root canal and mass of gutta-percha cones. The theoretical volume ratio of voids was influenced by the taper of canal, the size of apical preparation and the size of accessory gutta-percha cones. Greater apical preparation size and larger taper together with the use of smaller accessory cones reduced the volume ratio of voids in the apical third. The mathematical model provided a precise method to determine the theoretical volume ratio of voids in root-filled canals when using cold lateral compaction.

A Review of Tensors and Tensor Signal Processing

NASA Astrophysics Data System (ADS)

Cammoun, L.; Castaño-Moraga, C. A.; Muñoz-Moreno, E.; Sosa-Cabrera, D.; Acar, B.; Rodriguez-Florido, M. A.; Brun, A.; Knutsson, H.; Thiran, J. P.

Tensors have been broadly used in mathematics and physics, since they are a generalization of scalars or vectors and allow to represent more complex properties. In this chapter we present an overview of some tensor applications, especially those focused on the image processing field. From a mathematical point of view, a lot of work has been developed about tensor calculus, which obviously is more complex than scalar or vectorial calculus. Moreover, tensors can represent the metric of a vector space, which is very useful in the field of differential geometry. In physics, tensors have been used to describe several magnitudes, such as the strain or stress of materials. In solid mechanics, tensors are used to define the generalized Hooke’s law, where a fourth order tensor relates the strain and stress tensors. In fluid dynamics, the velocity gradient tensor provides information about the vorticity and the strain of the fluids. Also an electromagnetic tensor is defined, that simplifies the notation of the Maxwell equations. But tensors are not constrained to physics and mathematics. They have been used, for instance, in medical imaging, where we can highlight two applications: the diffusion tensor image, which represents how molecules diffuse inside the tissues and is broadly used for brain imaging; and the tensorial elastography, which computes the strain and vorticity tensor to analyze the tissues properties. Tensors have also been used in computer vision to provide information about the local structure or to define anisotropic image filters.

Descriptions of Free and Freeware Software in the Mathematics Teaching

NASA Astrophysics Data System (ADS)

Antunes de Macedo, Josue; Neves de Almeida, Samara; Voelzke, Marcos Rincon

2016-05-01

This paper presents the analysis and the cataloging of free and freeware mathematical software available on the internet, a brief explanation of them, and types of licenses for use in teaching and learning. The methodology is based on the qualitative research. Among the different types of software found, it stands out in algebra, the Winmat, that works with linear algebra, matrices and linear systems. In geometry, the GeoGebra, which can be used in the study of functions, plan and spatial geometry, algebra and calculus. For graphing, can quote the Graph and Graphequation. With Graphmatica software, it is possible to build various graphs of mathematical equations on the same screen, representing cartesian equations, inequalities, parametric among other functions. The Winplot allows the user to build graphics in two and three dimensions functions and mathematical equations. Thus, this work aims to present the teachers some free math software able to be used in the classroom.

Variational Approach in the Theory of Liquid-Crystal State

NASA Astrophysics Data System (ADS)

Gevorkyan, E. V.

2018-03-01

The variational calculus by Leonhard Euler is the basis for modern mathematics and theoretical physics. The efficiency of variational approach in statistical theory of liquid-crystal state and in general case in condensed state theory is shown. The developed approach in particular allows us to introduce correctly effective pair interactions and optimize the simple models of liquid crystals with help of realistic intermolecular potentials.