Presents responses to the problem appearing in the March, 2000 issue on data and graph interpretation. (ASK)

Finds that many activities that have been called "problems" are merely exercises or tricks. Offers a clearer definition of the word "problem". Notes some of the classroom implications of the new insight. (Author/CCM)

In this article we propose the following definition for advanced mathematical thinking: Thinking that requires deductive and rigorous reasoning about mathematical notions that are not entirely accessible to us through our five senses. We argue that this definition is not necessarily tied to a particular kind of educational experience; nor is it…

A mathematical problem is presented. Teachers are advised to spend time discussing the problem with their students, but they should avoid giving them too much guidance.

In a previous article in this series, it was suggested that it is part of our responsibility as teachers to attempt to induce "perturbations" in our students' mathematical thinking. Especially when teaching seniors and capable students at any level, it is important that we unsettle them, shake their perceptions and attempt, wherever…

The situations created by individual style of problem solving in mathematics by children are discussed. Children can solve their mathematical problem by correcting their misconceptions on their own with the use of open discussions with others.

The purpose of this paper was to investigate how it becomes possible for solvers to make drawings to advance their problem solving processes, in order to understand the use of drawings in mathematical problem solving more deeply. For this purpose, three examples in which drawings made by the solver played a critical role in the solutions have been…

Discusses a real-world problem-solving lesson that emerged when a high school math teacher used a motion detector with a CBL and graphing calculator to obtain the bounce data of a ping-pong ball. Describes the lesson in which students collect bad data then fill in the missing parabolas that result using critical components of parabolas and…

Describes the Six Circle problem which consists of the numbers 1-6, six circles, and asks whether it is possible to put the numbers in the circles--which are configured in a triangle--so that the sums of the three numbers on either side are the same. (NB)

The analysis of tautochrone problems involves the solution of integral equations. The paper shows how a reasonable assumption, based on experience with simple harmonic motion, allows one to greatly simplify such problems. Proposed solutions involve only mathematics available to students from first year calculus.

The Bernoulli numbers are among the most interesting and important number sequences in mathematics. They first appeared in the posthumous work "Ars Conjectandi" (1713) by Jacob Bernoulli (1654-1705) in connection with sums of powers of consecutive integers (Bernoulli, 1713; or Smith, 1959). Bernoulli numbers are particularly important in number…

The phenomenon of nonlinear resonance (sometimes called the "jump phenomenon") is examined and second-order van der Pol plane analysis is employed to indicate that this phenomenon is not a feature of the equation, but rather the result of accumulated round-off error, truncation error and algorithm error that distorts the true bounded solution onto…

We sometimes teach our students a method of finding all integral triples that satisfy the Pythagorean Theorem x[squared]+y[squared]=z[squared]. These are called Pythagorean triples. In this paper, we show how to solve the equation x[squared]+ky[squared]=z[squared], where again, all variables are integers.

This article discusses divisibility tests for any prime number.

The understanding and a liking towards mathematics can be very effectively developed in students by allowing them to find out the solutions for any basic problem or simulations, which are basically mathematical reenactments of nearly or completely hypothetical situations. The nontransitive relation of Efron's dice or the assignment of numbers in a…