When first learning how to write mathematical proofs, it is often easier for students to work with statements using the universal quantifier. Results that single out special cases might initially come across as more puzzling or even mysterious. In this article we explore three specific statements from abstract algebra that involve the number…

The "Reasoning Algebraically about Operations Casebook" was developed as the key resource for participants' Developing Mathematical Ideas seminar experience. The thirty-four cases, written by teachers describing real situations and actual student thinking in their classrooms, provide the basis of each session's investigation into the…

The method of proof by mathematical induction follows from Peano axiom 5. We give three properties which are often used in the proofs by mathematical induction. We show that these are equivalent. Supposing the well-ordering property we prove the validity of this method without using Peano axiom 5. Finally, we introduce the simplest form of…

Describes properties of Fibonacci numbers, including the law of recurrence and relationship with the Golden Ratio. Discussed are some applications of the numbers to sewage of towns on a river bank, resistances in electric circuits, and leafy stems in botany. Lists four references. (YP)

Describes a number sequence made by counting the occurrence of each digit from 9 to 0, catenating this count with the digit, and joining these numeric strings to form a new term. Presents a computer-aided proof and an analytic proof of the sequence; compares these two methods of proof. (YP)

Presents examples of 3-by-3 and 4-by-4 magic squares. Proves that the numbers 1 to 10 can not be fitted to the intersections of a pentagram and that the sum of the 4 numbers on each line is always 22. (YP)

A square-dance number is defined as an even number which has the property that the set which consisted of the numbers one through the even number can be partitioned into pairs so that the sum of each pair is a square. Theorems for identifying square-dance numbers are discussed. (YP)

Research on computational estimation and mental computation has received a considerable amount of attention from mathematics educators during the past decade. These proceedings resulted from a meeting to explore dimensions of number sense and its related fields. The participants came from three groups: mathematics educators actively pursuing…

This article discusses the classroom use of discovery of number pattern. Provided are examples of a table of squares, multiplications of numbers, and algebraic expressions. (YP)