The Stoltz-Cesaro Theorem, a discrete version of l'Hopital's rule, is applied to the summation of integer powers.

A. Lobb discovered an interesting generalization of Catalan's parenthesization problem, namely: Find the number L(n, m) of arrangements of n + m positive ones and n - m negative ones such that every partial sum is nonnegative, where 0 = m = n. This article uses Lobb's formula, L(n, m) = (2m + 1)/(n + m + 1) C(2n, n + m), where C is the usual…

Several jeeps are poised at base camp on the edge of a desert aiming to escort one of them as far as possible into the desert, while the others return to camp. They all have full tanks of gas and share their fuel to maximize penetration. In a friendly desert it is best to leave caches of fuel along the way to help returning jeeps. We solve the…

The "N" queens problem is a classic puzzle. It asks for an arrangement of "N" mutually non-attacking queens on an "N" x "N" chessboard. We discuss a recent variation called the "N" + "k" queens problem, where pawns are added to the chessboard to allow a greater number of non-attacking queens to be placed on it. We describe some of what is known…

The "box problem" from introductory calculus seeks to maximize the volume of a tray formed by folding a strictly rectangular sheet from which identical squares have been cut from each corner. In posing such questions, one would like to choose integral side-lengths for the sheet so that the excised squares have rational or integral side-length.…

A secretary has "n" letters and "n" addressed envelopes. Instead of matching each letter with the corresponding envelope, she inserts the letters in a random manner. What are the chances that every letter will be in the wrong envelope? In this article, the author presents a solution to this problem and discusses possible ways of placing the…

Teachers can use geometric patterns to promote students' understanding of functional relationships. In this article, the authors first look at a problem-solving process that supports the use of figural reasoning to explore and interpret geometric pattern tasks and generalize function rules. Second, the authors discuss a framework for…

This paper describes a framework for learning where learners undergo experimentations with the phenomena at hand according to progressive and staged goals. Bowling is used as a case study in this paper. The premise for experimentations is that learners can experience hidden efficacies, including the formation of "bad habits." A distinction is made…

These problems and activities have been used to demonstrate the uses of multiple representations as well as provide opportunities for students to practice using multiple representations in solving problems. (Contains 3 figures and 2 tables.)

This paper presents tips, activities, resources, and games that parents can use to help their children become more proficient in math. Some helpful tips offered are: (1) Be positive; (2) Play family games; (3) Avoid stereotypes; (4) Choose gifts that develop problem solving skills; (5) Expand your children's horizons; (6) Buy or borrow library…

Parents and teachers occasionally express concern that their children are too focused on the discovery of mathematics. The author remembers clearly one upset father who told him, "Just teach my daughter what she needs to know. Don't worry if she understands why it works. She just needs to know how to get the right answer." The author agrees with…

Traditional diagnosis treats behavior problems as "disorders" in the individual rather than "discord" in relationships. From an ecological perspective, a "problem" involves behavior that hurts self or others. Three global patterns of problems are: Inconsiderate of Others, Inconsiderate of Self, and Low Self-Worth. Strength-based assessment…

Basic and superior reasoning skills are woven into the clinical reasoning process just as they are used to solve any problem. As clinical reasoning is the central competence of medical education, development of these reasoning skills should occur throughout the undergraduate medical curriculum. The authors describe here a method of teaching…

In 2010, St. Leonards Primary School in Tasmania, along with other selected schools throughout Australia, trialled the draft Australian Curriculum: Mathematics. Mathematics had been a whole school focus at St. Leonards Primary School for several years, and the school found that the opportunity to be part of the trial strongly connected with their…

During the last decades of the twentieth century, changes in school curricula have resulted in proof-free or proof-lite curricula. In this article, the author argues that proof is, and should be seen to be, a central component in the school curriculum--from at least the middle of the primary years and upwards. He identifies proof with problem…