On flipping the classroom in large first year calculus courses

Over the course of two years, 2012--2014, we have implemented a ‘flipping’ the classroom approach in three of our large enrolment first year calculus courses: differential and integral calculus for scientists and engineers. In this article we describe the details of our particular approach and share with the reader some experiences of both instructors and students.     

Diagrams and math notation in e-learning: growing pains of a new generation

Current e-learning environments are ill-suited to college mathematics. Instructors/students struggle to post diagrams and math notation. A new generation of math-friendly e-learning tools, including WebEQ, bundled with Blackboard 6, and NetTutor's Whiteboard, address these problems. This paper compares these two systems using criteria for ideal math-friendly e-learning systems. NetTutor's Whiteboard is, apparently, the only system allowing two-way communication of both diagrams and math notation between instructor and students. This paper also summarizes a case study of two community college mathematics courses (calculus and algebra) using NetTutor over two semesters. Pilot studies, interviews and experimental problems revealed that NetTutor's Whiteboard is effective for 2-way communication of diagrams and math notation in college courses. Learning difficult concepts was comparable to face-to-face courses.     

An incentivized early remediation program in Calculus I

Strong prerequisite skills are essential to student success in the calculus sequence; however, many students arrive in Calculus I with weaknesses that are difficult for them to overcome. In this paper, we describe an approach to early incentivized remediation of prerequisite material in a Calculus I course. We present data that supports the idea that a lack of prerequisite knowledge is a significant hurdle for students, but also that participation in the remediation program is correlated with student success. In addition, the program allows for the very early identification of students at high risk of failing. The program is easy to implement, and it would be adaptable to a variety of other courses for which prerequisite knowledge is essential for success including science courses, engineering courses and other mathematics courses.     

Einstein’s equations and Clifford algebra

Einstein’s equations of the general theory of relativity are rewritten within a Clifford algebra. This algebra is otherwise isomorphic to a direct product of two quaternion algebras. A multivector calculus is developed within this Clifford algebra which differs from the corresponding complexified algebra used in the standard spacetime algebra approach.     

Maximum Principle for General Controlled Systems Driven by Fractional Brownian Motions

We obtain a maximum principle for stochastic control problem of general controlled stochastic differential systems driven by fractional Brownian motions (of Hurst parameter H>1/2). This maximum principle specifies a system of equations that the optimal control must satisfy (necessary condition for the optimal control). This system of equations consists of a backward stochastic differential equation driven by both fractional Brownian motions and the corresponding underlying standard Brownian motions. In addition to this backward equation, the maximum principle also involves the Malliavin derivatives. Our approach is to use conditioning and Malliavin calculus. To arrive at our maximum principle we need to develop some new results of stochastic analysis of the controlled systems driven by fractional Brownian motions via fractional calculus. Our approach of conditioning and Malliavin calculus is also applied to classical system driven by standard Brownian motions while the controller has only partial information. As a straightforward consequence, the classical maximum principle is also deduced in this more natural and simpler way.     

Teaching Probability in Intermediate Grades∗;†

This paper discusses three alternative models for structuring homework assignments in college level mathematics. A distributive model which incorporates both early and late review of previously learned material into the daily assignment schedule is explicated in detail. The relative efficacy of the conventional method for assigning homework and the distributive method in promoting achievement and favourable attitudes toward mathematics in a first semester course in calculus was investigated in a controlled experiment. Significant (p < 0.03) interactions between mathematical ability and assignment model were found on three of four unit tests. In each case, pupils with the weaker pre‐calculus background profited more from the distributive assignment schedule. There were no significant interactions or differences on a comprehensive post‐test or on attitudes toward mathematics. Implications for research and practice in college level mathematics, physics, and engineering courses are suggested.     

Classroom notes: Summing sequences having mixed signs

A result is discussed which permits the summing of series whose terms have more complicated sign patterns than simply alternating plus and minus. The Alternating Series Test, commonly taught in beginning calculus courses, is a corollary. This result, which is not difficult to prove, widens the series summable by beginning students and paves the way for understanding more advanced questions such as convergence of Fourier series. An elementary exposition is given of Dirichlet's Test for the convergence of a series and an elementary example suitable for a beginning calculus class and a more advanced example involving a Fourier series which is appropriate for an advanced calculus class are provided. Finally, two examples are discussed for which Dirichlet's Test does not apply and a general procedure is given for deciding the convergence or divergence of these and similar examples.     

Polynomial calculus: rethinking the role of calculus in high schools

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 their definition and exposition. We develop the beginning concepts of differential and integral calculus using only concepts and skills found in secondary algebra and geometry. It is our underlining objective to strengthen students' knowledge of these topics in an effort to prepare them for advanced mathematics study. The purpose of this reconstruction is not to alter the teaching of limit-based calculus but rather to affect students' learning and understanding of mathematics in general by introducing key concepts during secondary mathematics courses. This approach holds the promise of strengthening more students' understanding of limit-based calculus and enhancing their potential for success in post-secondary mathematics.     

Collaborative Workshops and Student Academic Performance in Introductory College Mathematics Courses: A Study of a Treisman Model Math Excel Program

High failure rates in introductory college mathematics courses, particularly among underrepresented groups of students, have been of concern for many years. One approach to the problem experiencing some success has been Treisman's Emerging Scholars workshop model. The model involves supplemental workshops in which students solve problems in collaborative learning groups. This study reports on the effectiveness of Math Excel, an implementation of the Treisman model for introductory mathematics courses (college algebra, precalculus, differential calculus, and integral calculus) at Oregon State University over five academic terms. Regression analyses revealed a significant effect on achievement (.671 grade points on a 4‐point scale) favoring Math Excel students. Even after adjusting for prior mathematics achievement using linear regression with SAT‐M as predictor, Math Excel groups' grade averages were over half a grade point better than predicted (significant at the .001 level). This study provides supporting evidence that programs like Math Excel can help students in making a successful transition to college mathematics study.     

The notion of motion: covariational reasoning and the limit concept

This paper investigates outcomes of building students’ intuitive understanding of a limit as a function's predicted value by examining introductory calculus students’ conceptions of limit both before and after instruction. Students’ responses suggest that while this approach is successful at reducing the common limit equals function value misconception of a limit, new misconceptions emerged in students’ responses. Analysis of students’ reasoning indicates a lack of covariational reasoning that coordinates changes in both x and y may be at the root of the emerging limit reached near x = c misconception. These results suggest that although dynamic interpretations of limit may be intuitive for many students, care must be taken to foster a dynamic conception that is both useful at the introductory calculus level and is in line with the formal notion of limit learned in advanced mathematics. In light of the findings, suggestions for adapting the pedagogical approach used in this study are provided.     

Improving student learning in calculus through applications

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.     

Analysis of projectile motion in view of fractional calculus

The projectile motion is examined by means of the fractional calculus. The fractional differential equations of the projectile motion are introduced by generalizing Newton’s second law and Caputo’s fractional derivative is considered to use the physical initial conditions. In the absence of air resistance it is found that at certain conditions, the range and the maximum height of the projectile obtained by using the fractional calculus give the same results of the classical calculus. It is also found that, unlike the classical projectile motion, the launching angle that maximizes the horizontal range is a function of the arbitrary order of the fractional derivative α. Moreover, in a resistant medium, the parametric equations are expressed in terms of Mittag-Leffler function and the results agree with those of the classical projectile as α → 2. Moreover, the trajectories of the projectile are discussed in graphs and compared with those of the classical calculus. In order to explore the validity of modelling the projectile motion by the fractional approach, we compared our results with the experimental data of mortar.     

A fuzzy logic for an ordinal sum t-norm

Among the class of residuated fuzzy logics, a few of them have been shown to have standard completeness both for propositional and predicate calculus, like Gödel, NM and monoidal t-norm-based logic systems. In this paper, a new residuated logic NMG, which aims at capturing the tautologies of a class of ordinal sum t-norms and their residua, is introduced and its standard completeness both for propositional calculus and for predicate calculus are proved.     

Stability Properties for Feynman’s Operational Calculus in the Combined Continuous/Discrete Setting

We establish some stability theorems for Feynman’s operational calculus in the setting where the time-ordering measures are allowed to have both continuous and discrete parts. In particular, we investigate stability in a number of special cases of this blended approach to the operational calculus.Mathematics Subject Classifications (2000) Primary: 46J15, 47A56, 47A60, 60B10; secondary: 46N50, 47N50.     

Hahn Difference Operator and Associated Jackson–Nörlund Integrals

This paper is devoted for a rigorous investigation of Hahn’s difference operator and the associated calculus. Hahn’s difference operator generalizes both the difference operator and Jackson’s q-difference operator. Unlike these two operators, the calculus associated with Hahn’s difference operator receives no attention. In particular, its right inverse has not been constructed before. We aim to establish a calculus of differences based on Hahn’s difference operator. We construct a right inverse of Hahn’s operator and study some of its properties. This inverse also generalizes both Nörlund sums and the Jackson q-integrals. We also define families of corresponding exponential and trigonometric functions which satisfy first and second order difference equations, respectively.     

The Effects of Non‐CAS Graphing Calculators on Student Achievement and Attitude Levels in Mathematics: A Meta‐Analysis

Forty‐two studies comparing students with access to graphing calculators during instruction to students who did not have access to graphing calculators during instruction are the subject of this meta‐analysis. The results on the achievement and attitude levels of students are presented. The studies evaluated cover middle and high school mathematics courses, as well as college courses through first semester calculus. When calculators were part of instruction but not testing, students' benefited from using calculators while developing the skills necessary to understand mathematics concepts. When calculators were included in testing and instruction, the procedural, conceptual, and overall achievement skills of students improved.     

On Metric and Calmness Qualification Conditions in Subdifferential Calculus

The paper contains two groups of results. The first are criteria for calmness/subregularity for set-valued mappings between finite-dimensional spaces. We give a new sufficient condition whose subregularity part has the same form as the coderivative criterion for “full” metric regularity but involves a different type of coderivative which is introduced in the paper. We also show that the condition is necessary for mappings with convex graphs. The second group of results deals with the basic calculus rules of nonsmooth subdifferential calculus. For each of the rules we state two qualification conditions: one in terms of calmness/subregularity of certain set-valued mappings and the other as a metric estimate (not necessarily directly associated with aforementioned calmness/subregularity property). The conditions are shown to be weaker than the standard Mordukhovich–Rockafellar subdifferential qualification condition; in particular they cover the cases of convex polyhedral set-valued mappings and, more generally, mappings with semi-linear graphs. Relative strength of the conditions is thoroughly analyzed. We also show, for each of the calculus rules, that the standard qualification conditions are equivalent to “full” metric regularity of precisely the same mappings that are involved in the subregularity version of our calmness/subregularity condition. The research of Jiří V. Outrata was supported by the grant A 107 5402 of the Grant Agency of the Academy of Sciences of the Czech Republic.     

Parking buses in a depot with stochastic arrival times

Given buses of different types arriving at a depot during the evening, the bus parking problem consists of assigning these buses to parking slots in such a way that they can be dispatched adequately to the next morning routes without moving them between their arrivals and departures. In practice, the bus arrival times deviate stochastically from the planned schedule. In this paper, we introduce for this problem two solution approaches that produce solutions which are robust to variations in the arrival times. The first approach considers that each arrival can deviate from its planned arrival order (sooner or later) by at most k positions, where k is a predefined parameter. In the second approach, the objective aims at minimizing the expectation of a function positively correlated with the number of buses that make the planned solution infeasible because they arrive too late or too early. In both approaches, the problem is modeled as an integer linear program that can be solved by a commercial mip solver. Computational results obtained on instances derived from a real-world dataset are reported.     

Reproducibility of linear and nonlinear input-output systems

An algebraic setting for the Roman-Rota umbral calculus is introduced. It is shown how many of the umbral calculus results follow simply by introducing a comultiplication map and requiring it to be an algebra map. The same approach is used to construct a q-umbral calculus. Our umbral calculus yields some of Andrews recent results on Eulerian families of polynomials as corollaries. The homogeneous Eulerian families are studied. Operator and functional expansions are also included.     

Discussions on Newton's infinitesimals in eighteenth-century Anglo-America

The controversy in England over Newton's fluxionary calculus following the publication in 1734 of Bishop George Berkeley's The Analyst was reflected in the correspondence between Cadwallader Colden of New York and his friends in the middle 1740s. Colden wrote “An Introduction to the Doctrine of Fluxions” after reading The Analyst, and it was the occasion for the discussions that followed. His friends either doubted the value of the calculus and the validity of infinitesimals, or were noncommittal. Colden's essay was the only published defense of Newton's calculus by a colonist in eighteenth-century Anglo-America. There was a lack of interest in the calculus in both Great Britain and America until well into the nineteenth century. In the following we suggest reasons for that lack of interest.