David M. Bressoud

• Macalester College

In the United States, the first course in single-variable calculus is considered tertiary level mathematics. Initially offered in high schools as a means for strong students to do college-level work, it is now taken by over 20% of high school students and perceived to be a prerequisite for admission into selective colleges and universities. This ar...

For far too many students, Real Analysis is a dreaded course that proceeds from unmotivated definitions to formal and impenetrable theorems with little sense of why the course unfolds as it does. This article describes the experience of a radically different approach that drew on the history of mathematics to confront students with the uncertaintie...

This is a survey of the developments in the first two years of undergraduate mathematics, beginning in the early 1950s and continuing up to the present. It documents the repeated efforts at making this instruction relevant to the partner disciplines, especially Biology, and describes the challenges for the future.

We present findings from a recently completed census survey of all mathematics departments in the United States that offer a graduate degree in mathematics. The census survey is part of a larger project investigating institutional features that influence student success in the introductory mathematics courses that are required of most STEM majors i...

The diversity of the research in the field of Calculus education makes it difficult to produce an exhaustive state-of-the-art summary. We therefore focus on the main trends in the field in order to detect punctual evolutions that permit us to go beyond this survey and to put forward new research questions. The research results, in spite of their va...

This survey focuses on the main trends in the field of calculus education. Despite their variety, the findings reveal a cornerstone issue that is strongly linked to the formalism of calculus concepts and to the difficulties it generates in the learning and teaching process. As a complement to the main text, an extended bibliography with some of the...

Students who succeed in high school calculus become discouraged and quit in college. Why?

In this study, we developed a three-dimensional framework to classify post-secondary Calculus I final exams. Our Exam Characterization Framework (ECF) classifies individual exam items according to the cognitive demand required to answer the item, the representational context in which the item is asked, and the format of the item. Our results from u...

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 towar...

In these days of tight budgets and pressure to improve retention rates for science and engineering majors, many mathematics departments want to know what works, what are the most productive means of improving the effectiveness of calculus instruction. This was the impetus behind the study of Characteristics of Successful Programs in College Calculu...

In fall 2010, the Mathematical Association of America undertook the first large-scale study of postsecondary Calculus I instruction in the United States, employing multiple instruments. This report describes this study, the background of the students who take calculus and changes from the start to the end of the course in student attitudes towards...

This article explores the history of the Fundamental Theorem of Integral Calculus, from its origins in the 17th century through its formalization in the 19th century to its presentation in 20th century textbooks, and draws conclusions about what this historical development tells us about how to teach this fundamental insight of calculus.

Returning to the beginnings of trigonometry—the circle—has implications for how we teach it.

This paper illustrates some of the power and beauty of determinant evaluations, beginning with Cauchy’s proof of the Vandermonde determinant evaluation, continuing through the Weyl denominator formulas and some open conjectures on alternating-signmatrices, and ending with the Izergin-Korepin determinant expansion [cf. V. E. Korepin, N. M. Bogoliubo...

In November 2005, the faculty of Macalester College voted to institute a graduation requirement in Quantitative Thinking (QT) that is truly interdisciplinary. It currently draws on courses from thirteen departments including Anthropology, Economics, Geography, Political Science, Theater, Mathematics, Environmental Science, and Geology. This article...

Over the past decade mathematicians have been increasingly concerned about the number of undergraduate students studying mathematics and the consequences thereof, an important one being a possible decreasing need for mathematics faculties to teach them. The author reviews current data, finding cause for both optimism, in increasing numbers, and pes...

In November 2005, the faculty of Macalester College voted to institute a graduation requirement in Quantitative Thinking (QT) that is truly interdisciplinary. It currently draws on courses from thirteen departments including Anthropology, Economics, Geography, Political Science, Theater, Mathematics, Environmental Science, and Geology. This article...

Without Abstract

This is an exposition and elaboration on work of W.H. Burge which demonstrates the connections among the various combinatorial interpretations of the multiple summations which arise in generalizations of the Rogers-Ramanujan identities. It includes some new results on partitions with restrictions on the succesive ranks and an extension of the Roger...

What is quantitative literacy? How do you teach it? How do you measure it? How can you develop a program that will ensure that all undergraduates have it by the time they graduate? During the academic year 2001–02, faculty from Macalester College wrestled with these questions and found answers. These have led to a pilot program, Quantitative Method...

The Mathematical Association of America's Committee on the Undergraduate Program in Mathematics (CUPM) is charged with making recommendations to guide mathematical sciences departments in designing undergraduate curricula. "Undergraduate Programs and Courses in the Mathematical Sciences: CUPM Curriculum Guide, 2004" is based on four years of work,...

This paper highlights three known identities, each of which involves sums over alternating sign matrices. While proofs of all three are known, the only known derivations are as corollaries of difficult results. The simplicity and natural combinatorial interpretation of these identities, however, suggest that there should be direct, bijective proofs...

This is an overview of some of the changes that have been occurring in undergraduate mathematics education. It is based on a talk given at a workshop held by the Mathematical Association of America to determine what chemists expect their students to learn from supporting undergraduate mathematics courses in terms of understanding, content, and use...

Versions of Bailey''s lemma which change the base from q to q 2 or q 3 are given. Iterates of these versions give many new versions of multisum Rogers-Ramanujan identities.

We present three infinite families of partitions that, for corresponding parameters, are equinumerous. This generalizes an identity proved by the first author (Pacific Journal of Mathematics, 77(1) (1978)) and two identities proved by the second and third authors (Relatrio Tcnico \textn\texto {\text{n}}\underline {\text{o}} 08/99 IMECC-UNICAMP,...

This paper highlights three known identities, each of which involves sums over alternating sign matrices. While proofs of all three are known, the only known derivations are as corollaries of difficult results. The simplicity and natural combinatorial interpretation of these identities, however, suggest that there should be direct, bijective proofs...

We use elementary methods to prove formulas that represent sums of restricted classes of Schur functions as ratios of determinants. This includes recent formulas for sums over bounded partitions with even parts and sums over bounded partitions whose conjugates have only even parts. All of these formulas imply plane partition generating functions.

Versions of Bailey's lemma which change the base from q to q^2 or q^3 are given. Iterates of these versions give many new versions of multisum Rogers-Ramanujan identities. We also prove Melzer's conjectures for the Fermionic forms of the supersymmetric analogues of Virasoro characters.

This is an introduction to recent developments in algebraic combinatorics and an illustration of how research in mathematics actually progresses. The author recounts the story of the search for and discovery of a proof of a formula conjectured in the late 1970s: the number of n x n alternating sign matrices, objects that generalize permutation matr...

Introduction Perusing the four volumes of Muir's The Theory of Determinants in the Historical Order of Development, one might be tempted to conclude that the theory of determinants was well and truly beaten to death in the nineteenth century. In fact, the field is thriving, and it has continued to yield challenging problems of deceptive elegance an...

We use elementary methods to prove product formulas for sums of restricted classes of Schur functions. These imply known identities for the generating function for symmetric plane partitions with even column height and for the generating function for symmetric plane partitions with an even number of angles at each level.

Elementary proofs are given for sums of Schur functions over partitions into at most n parts each less than or equal to m for which i) all parts are even, ii) all parts of the conjugate partition are even. Also, an elementary proof of a recent result of Ishikawa and Wakayama is given.

: The number of n n matrices whose entries are either -1, 0, or 1, whose row- and column- sums are all 1, and such that in every row and every column the non-zero entries alternate in sign, is proved to be [1!4! ...(3n- 2)!]/[n!(n +1)!...(2n- 1)!], as conjectured by Mills, Robbins, and Rumsey. 1 original version written December 1992. The Maple pac...

Peter Borwein has conjectured that certain polynomials have non-negative coe#cients. In this paper we look at some generalizations of this conjecture and observe how they relate to the study of generating functions for partitions with prescribed hook di#erences. A combinatorial proof of the generating function for partitions with prescribed hook di...

Peter Borwein has conjectured that certain polynomials have non-negative coefficients. In this paper we look at some generalizations of this conjecture and observe how they relate to the study of generating functions for partitions with prescribed hook differences. A combinatorial proof of the generating function for partitions with prescribed hook...

Two stones build two houses. Three build six houses. Four build four and twenty houses. Five build hundred and twenty houses. Six build Seven hundreds and twenty houses. Seven build five thousands and forty houses. From now on, [exit and] ponder what the mouth cannot speak and the ear cannot hear. (Sepher Yetsira IV,12) Abstract: The number of n ×...

Major Percy A. MacMahon's first paper on plane partitions [4] included a conjectured generating function for symmetric plane partitions. This conjecture was proven almost simultaneously by George Andrews and Ian Macdonald, Andrews using the machinery of basic hypergeometric series [1] and Macdonald employing his knowledge of symmetric functions [3]...

Major Percy A. MacMahon's first paper on plane partitions included a conjectured generating function for symmetric plane partitions. This conjecture was proven almost simultaneously by George Andrews and Ian Macdonald, Andrews using the machinery of basic hypergeometric series and Macdonald employing his knowledge of symmetric functions. The purpos...

. There are four values of s for which the hypergeometric function 2 F 1 ( 1 2 Gamma s; 1 2 + s; 1; Delta) can be parametrized in terms of modular forms; namely, s = 0, 1 3 , 1 4 , 1 6 . For the classical s = 0 case, the parametrization is in terms of the Jacobian theta functions ` 3 (q), ` 4 (q) and is related to the arithmetic-geometric mean iter...

We give a unified exposition of the use of the Gessel-Viennot interpretation of determinants as n-tuples of lattice paths to demonstrate the combinatorial equivalence of four classical definitions of the Schur function. This includes a combinatorial proof of the Jacobi- Trudi identity.

We translate Goulden’s combinatorial proof of the Jacobi-Trudi identity into the language of lattice paths and then use the Gessel-Viennot technique to prove a general identity between a determinant involving complete symmetric functions and a sum of skew Schur functions.

Unimodality of the Gaussian polynomial is proved by establishing an identity for these polynomials from which an inductive proof of unimodality follows.

Let p be an odd prime, ζ a primitive pth root of unity. It is proved that Π(1 + iζk)k, 1 ≤ k ≤ p − 1 is a perfect pth power in (iζ) only if p divides Σtp − 3, , and that it is a perfect pth power for P = 241. Properties of the general product Π(1 + aqk)k, 1 ≤ k ≤ n, are investigated.

There is a curious contradiction in the standard presentation of integral calculus. It is an ancient subject rooted in the geometric investigations of Archimedes of Syracuse (287 –212 B.C.).

Consider a level surface in three dimensions,f(x,y,z)=c.

We interrupt our study of differential calculus to apply the results of Section 4 of Chapter 7 to the outstanding problem of evaluating pullbacks.

Much as Newton’s contribution to calculus was less in the discovery of the techniques than in his vision of how they linked together and what could be done with them, so the equations of James Clerk Maxwell (1831–1879) were not, with one partial exception, his discovery, yet he placed his mark upon them in recognizing their basic unity and what the...

With this chapter we begin the study of functions whose domain and range consist of several variables.

While the terminology of calculus that we have at hand is certainly sufficient to prove the converse of Theorem 1.3, namely, that Newton’s law of gravity implies that planets must move in elliptical orbits with the sun at one focus, this and other arguments we are to make will be greatly simplified if we adopt a language developed in the late ninet...

We recall that a vector field, \( \overrightarrow F \) is differentiable at \(\vec{c}\) if and only if there exists a linear transformation, \(\vec L_c \).

We are not quite ready to prove that Newton’s law of gravitational attraction implies Kepler’s second law. We need to take a closer look at the Calculus of vector functions in the light of the vector algebra described in the last chapter.

For scalar functions of one variable, the fundamental theorem of calculus is a powerful tool for integration. It says that there exists an associated function, often called an antiderivative, such that the integral of the scalar function can be evaluated by looking at the antiderivative at the endpoints of the interval. Specifically, it is the foll...

A linear transformation is a function,\( \vec{L}, \) from one or more real variables to one or more real variables, that satisfies the following two conditions for any values of \( \vec{a} \) and \( \vec{b} \) in the domain and any real constant c

Second Year Calculus: From Celestial Mechanics to Special Relativity covers multi-variable and vector calculus, emphasizing the historical physical problems which gave rise to the concepts of calculus. The book carries us from the birth of the mechanized view of the world in Isaac Newton's Mathematical Principles of Natural Philosophy in which math...

We present ten assorted problems which have arisen in the attempt to understand the Z-statistic, a statistic on words which plays a crucial role in the proof of Andrews' q-Dyson conjecture.

Certain basic hypergeometric series with multiple indices of summation are interpreted as generating functions for weighted lattice paths. The approach uses ideas of William Surge and gives rise to identities analogous to the Rogers-Ramanujan identities.

This paper presents a proof and investigation of a curious identity which is implicit in work of K. O’Hara [7] and which was extracted and first explicitly stated by D. Zeilberger [8].