The November issue of The College Mathematics Journal deliberately highlights Classroom Capsules. Instead of only one or two of these short notes on topics of immediate classroom relevance, we have here fully 10 Capsules on, for instance, the singular value decomposition of a matrix, finding the center of mass of a soft spring, error in Euler's method, applying the Hölder inequality to the Cobb-Douglas production function, and using Abel's theorem on the Wronskian to solve differential equations. In particular, don't miss Herb Bailey and Dan Kalman on ‘Walking with a Slower Friend.'
The November issue also includes the delightful ‘How to be a Good Teacher is an Undecidable Problem' by Erica Flapan, this year's recipient of the Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching. And another highlight is the book review (by Samuel Goldberg) of Willard Wells' ‘Apocalypse When?'.
2010 Haimo Award Lecture
How to be a good teacher is an undecidable problem
In this article, I describe my attempts to use various pedagogical techniques to teach college level mathematics and the particular problems that I ran into with each. I explain how I reached the conclusion that there is no best pedagogical approach. Rather each person's teaching methods should fit their personality, their mathematical preferences, and the needs and goals of their courses and their students. MSC: 97D40, 99D99
One Problem, Nine Student-Produced Proofs
Geoffrey Birky, Connie Campbell, Jim Sandefur, Kay Somers, Manya Raman
This paper tells the story of what happened when students in the authors' sophomore-level introduction-to-proof classes were given a theorem to prove with no expectation about what proof method to use. The paper discusses the nine different student-produced proofs of the statement: If n is an integer such that n = 3, then n3 > (n + 1)2. MSC: 97E50
Easiest Lights Out Games
The game Lights Out and its mathematical predecessor, the sigma-plus game, has inspired an extensive mathematical literature. In this paper, the original game and a borderless version played on a torus are considered. We define an easy game to be one in which pushing the buttons that are originally lit solves the game. Easy games are classified for all square boards and a strategy is introduced which, for certain board sizes, will readily solve any Lights Out puzzle. MSC: 97A20, 00A08, 15B33
Introducing Czedli-type Islands
Eszter K. Horvath, Attila Mader, Andreja Tepavcevic
The notion of an island has surfaced in recent algebra and coding theory research. Discrete versions provide interesting combinatorial problems. This paper presents the one-dimensional case with finitely many heights, a topic convenient for student research. MSC: 05A99, 05C05, 06A07, 06B99
Derivative Sign Patterns
Analysis of the patterns of signs of infinitely differentiable real functions shows that only four patterns are possible if the function is required to exhibit the pattern at all points in its domain and that domain is the set of all real numbers. On the other hand all patterns are possible if the domain is a bounded open interval. MSC: 26A24, 26A06 sion.
Limit Interchange and L'Hôpital's Rule
Conventional application of these two calculus staples is stretched here, somewhat recreationally, but also to raise solid questions about the role of limit interchange in analysis - without, however, delving any deeper than first-year Calculus. MSC: 00-01, 00A05, 00A08, and 00A09
Walking with a Slower Friend
Herb Bailey and Dan Kalman
Fay and Sam go for a walk. Sam walks along the left side of the street while Fay, who walks faster, starts with Sam but walks to a point on the right side of the street and then returns to meet Sam to complete one segment of their journey. We determine Fay's optimal path minimizing segment length, and thus maximizing the number of times they meet during the walk. Two solutions are given: one uses derivatives; the other uses only continuity. MSC: 97I40, 00A69
Cobb-Douglas and Hölder
Holder's inequality is here applied to the Cobb-Douglas production function to provide simple estimates to total production. MSC: 26D15, 91B38
The Center of Mass of a Soft Spring
Amitabh Joshi and Juan D. Serna
This article uses calculus to find the center of mass of a soft, vertically suspended, cylindrical helical spring, which necessarily is stretched non-uniformly by the action of gravity. A general expression for the vertical position of the center of mass is obtained and compared with other results in the literature. MSC: 00A69, 00A79, 74A99
An Intuitive Proof of the Singular Value Decomposition of a Matrix
Using a simple trigonometric limit, we provide an intuitive geometric proof of the Singular Value Decomposition of an arbitrary matrix. MSC: 15A18, 15A23
Discretization and Rounding Error in Euler’s Method
Euler's method for solving initial value problems is an excellent vehicle for observing the relationship between discretization error and rounding error in numerical computation. Reductions in stepsize, in order to decrease discretization error, necessarily increase the number of steps and so introduce additional rounding error. The problem is common and can be quite troublesome. We examine here a simple device, well known to those versed in the fixed point computations employed many years ago, that can help delay the onset of this problem.
MSC: 65-01, 65-04, 65L05
Abel’s Theorem Simplifies Reduction of Order
We give an alternative to the standard method of reduction or order, in which one uses one solution of a homogeneous, linear, second order differential equation to find a second, linearly independent solution. Our method, based on Abel's Theorem, is shorter, less complex and extends to higher order equations.
MSC: 34A05, 34-01
Averaging Sums of Powers of Integers
When is the average of sums of powers of integers itself a sum of the first n integers raised to a power? We provide all solutions when averaging two sums, and provide some conditions regarding when larger averages may have solutions. MSC: 39B05, 00A08
Uncountably Generated Ideals of Functions
Maximal ideals in the ring of continuous functions on the closed interval [0,1] are not finitely generated. This is well-known. What is not as well-known, but perhaps should be, is the fact that these ideals are not countably generated although the proof is not harder! We prove this here and use the result to produce some non-prime ideals in the ring of continuous functions on the open interval (0,1) which also cannot be countably generated.
MSC: 13A15, 13E15, 46J20
PROBLEMS AND SOLUTIONS
Apocalypse When? by Willard Wells
Reviewed by Samuel Goldberg