Friday, May 22, 2015

New Textbook: An Invitation to Real Analysis

An Invitation to Real Analysis
by Luis F. Moreno
List: $75.00
MAA Member: $60.00

An Invitation to Real Analysis is written both as a stepping stone to higher calculus and analysis courses, and as foundation for deeper reasoning in applied mathematics. This book also provides a broader foundation in real analysis than is typical for future teachers of secondary mathematics. In connection with this, within the chapters, students are pointed to numerous articles from The College Mathematics Journal and The American Mathematical Monthly. These articles are inviting in their level of exposition and their wide-ranging content.

Axioms are presented with an emphasis on the distinguishing characteristics that new ones bring, culminating with the axioms that define the reals. Set theory is another theme found in this book, beginning with what students are familiar with from basic calculus. This theme runs underneath the rigorous development of functions, sequences, and series, and then ends with a chapter on transfinite cardinal numbers and with chapters on basic point-set topology.

Differentiation and integration are developed with the standard level of rigor, but always with the goal of forming a firm foundation for the student who desires to pursue deeper study. A historical theme interweaves throughout the book, with many quotes and accounts of interest to all readers.

Over 600 exercises and dozens of figures help the learning process. Several topics (continued fractions, for example), are included in the appendices as enrichment material. An annotated bibliography is included.

Interested in an examination copy? Learn more here.

Friday, May 8, 2015

Now in Print! The Heart of Calculus

The Heart of Calculus: Explorations and Applications
Philip M. Anselone and John W. Lee
List Price: $60.00 | MAA Member Price: $48.00

This book contains enrichment material for courses in first and second year calculus, differential equations, modeling, and introductory real analysis. It targets talented students who seek a deeper understanding of calculus and its applications. The book can be used in honors courses, undergraduate seminars, independent study, capstone courses taking a fresh look at calculus, and summer enrichment programs. The book develops topics from novel and/or unifying perspectives. Hence, it is also a valuable resource for graduate teaching assistants developing their academic and pedagogical skills and for seasoned veterans who appreciate fresh perspectives.

The explorations, problems, and projects in the book impart a deeper understanding of and facility with the mathematical reasoning that lies at the heart of calculus and conveys something of its beauty and depth. A high level of rigor is maintained. However, with few exceptions, proofs depend only on tools from calculus and earlier. Analytical arguments are carefully structured to avoid epsilons and deltas. Geometric and/or physical reasoning motivates challenging analytical discussions. Consequently, the presentation is friendly and accessible to students at various levels of mathematical maturity. Logical reasoning skills at the level of proof in Euclidean geometry suffice for a productive use of the book.

There are 16 chapters in the book, divided about equally between pure and applied mathematics. The first three chapters are on fundamentals of differential calculus and the last three are on the monumental discoveries of Newton and Kepler on celestial motion and gravitation. The intervening chapters present significant topics in pure and applied mathematics chosen for their intrinsic interest, historical influence, and continuing importance. There is great flexibility in the choice of which chapters to cover and the order of coverage because chapters are essentially independent of each other.

Friday, April 24, 2015

MAA Books Beat: New Calculus Textbook

by Steve Kennedy, MAA Senior Acquisitions Editor

What do you do at your school with a first–year student who scored a four or five in AP calculus in high school? There is, at least at Carleton, an awkward articulation between the coverage of the AP AB syllabus and our first-year calculus sequence.

For the first several weeks of our Calculus II class we are covering material that is, at least partially, in an AP calculus class: methods of integration; applications of the integral, including arc length and surface area; some elementary differential equations. But students who take Calculus I at Carleton haven’t seen any of this. It makes for a strange dynamic in the classroom: For some students everything is new and confusing; for others (the AP students) large parts are review. The former feel at a disadvantage, the latter a little bored.

We don’t really want to have two strands through our calculus sequence, one for students with AP experience and another for students who take all their calculus at Carleton. Several years ago we took the AP-experienced students and gave them a half-term course in modeling while the non-AP students took our standard Calculus II material. Then the two groups came together for the second half of the term to study sequences and series together. This turned out to be logistically difficult (and confusing to folks in other departments of the college), and we abandoned the experiment after a few years.

Friday, March 27, 2015

MAA Books Store: Buy 1 Get One 1/2 Off

Buy I, Mathematician and get 50% off* A Mathematician Comes of Age.

Hurry! This deal ends Tuesday, March 31.

I, Mathematician
Peter Casazza, Steven G. Krantz and Randi D. Ruden, Editors
List: $50.00 MAA Member: $40.00

Mathematicians have pondered the psychology of the members of our tribe probably since mathematics was invented, but for certain since Hadamard's The Psychology of Invention in the Mathematical Field. The editors asked two dozen prominent mathematicians (and one spouse thereof) to ruminate on what makes us different. The answers they got are thoughtful, interesting and thought-provoking.

A Mathematician Comes of Age
Steven G. Krantz
List: $60.00 MAA Member: $48.00

The book provides background, data, and analysis for understanding the concept of mathematical maturity. It turns the idea of mathematical maturity from a topic for coffee-room conversation to a topic for analysis and serious consideration.

*Discount is off the list price. No code needed. To receive the discount, both items must be placed in your cart. Offer expires March 31, 2015 at midnight.

Friday, March 13, 2015

New MAA Textbook: Modeling Approach in Biology

by Steve Kennedy

When biology was primarily a descriptive science, biologists had little need of mathematical services (and our science). But a revolution has occurred—more precisely, is occurring—in biology, and the discipline is becoming increasingly mathematical and computational. In 2003, a report (Bio2010) from the National Academy of Sciences (NAS) set a goal of having biology undergraduates be quantitatively literate by 2010:

It is essential that biology undergraduates become quantitatively literate, studying the mathematical concepts of change, modeling, equilibria and stability, structure of a system, interactions among components, data and measurement, visualization, and algorithms. Every student should acquire the ability to analyze issues in these contexts in some depth, using analytical methods (e.g., pencil and paper) and appropriate computational tools. An appropriate course of study would include aspects of probability, statistics, discrete models, linear algebra, calculus and differential equations, modeling and programming.

Five years past 2010, the goal hasn't been met fully. The current state of affairs is well described in the draft CUPM Curriculum Guide. That report, besides enumerating biological and mathematical core competencies for biomathematics study, strongly urges foundational courses in modeling and data analysis as the beginning steps of such study.

And that brings me, finally, to my point. MAA Books has just released a book perfectly suited for that foundational modeling course, Jim Cornette and Ralph Ackerman’s Calculus for the life Sciences: A Modeling Approach. If you teach a life sciences calculus course, you may already be aware of the existence of this book, as draft versions have been on the Internet for several years. Reading this terrific and innovative book, you realize that it could have been written in reaction to the first sentence of the NAS report—it addresses, in a deep and significant way, every single mathematical concept listed in the quote above.

Chapter 1 is a primer on modeling in the biological realm, and biological modeling is the theme and frame for the entire book. The authors build models of bacterial growth, light penetration through a column of water, and dynamics of a colony of mold in the first few pages. In each case there is actual data that needs fitting. In the case of the mold colony, that data is a set of photographs of the colony growing on a ruled sheet of graph paper, and the students need to make their own approximations. Fundamental questions about the nature of mathematical modeling—trying to approximate a real-world phenomenon with an equation—are all laid out for the students to wrestle with.

Students intending to major in the life sciences now outnumber engineering and physical science majors in college and university Calculus I courses. Cornette and Ackerman’s textbook takes into account the needs of that plurality. And it does so not just by including examples and problems with a biological flavor—the authors have reimagined the entire year-long calculus course with the needs of biology students as the organizing principle. Thus, to mention just one example among many, your students using this book could get a substantial dose of discrete and continuous dynamical systems theory in their year of calculus study.

Cornette and Ackerman have produced a beautifully written introduction to the uses of mathematics in the life sciences. The exposition is crystalline, the problems are overwhelmingly from biology and interesting and rich, and the emphasis on modeling is invigorating. This book should become the standard text for this course. Please take special note of the student-friendly price—$35 for an ebook.

This article was written for MAA FOCUS as part of MAA Books Beat. It appears in the February/March 2015 issue.