Mathematics teachers need to teach students more than just a laundry list of mathematical skills. Many of us also want to train our students to use mathematical habits of mind. However, traditional grading schemes usually do not allow us to reward intellectual courage and modesty. In this paper, I report on my implementation of specifications…

This article describes the technique of introducing a new variable in some calculus problems to help students master the skills of integration and evaluation of limits. This technique is algorithmic and easy to apply.

The necessity of using inquiry-based learning (IBL) was recently recommended by studies and reports made for the European Commission. Several European projects are devoted to the widespread use of IBL methods. The effects of using IBL are studied worldwide. In the framework of the Seventh Framework Program (FP7) project PRIMAS, a series of…

Bayes's theorem is notorious for being a difficult topic to learn and to teach. Problems involving Bayes's theorem (either implicitly or explicitly) generally involve calculations based on two or more given probabilities and their complements. Further, a correct solution depends on students' ability to interpret the problem correctly. Most people…

The National Council of Teachers of Mathematics calls for an increased emphasis on proof and reasoning in school mathematics curricula. Given such an emphasis, mathematics teachers must be prepared to structure curricular experiences so that students develop an appreciation for both the value of proof and for those strategies that will assist them…

In its mathematics standards, National Council of Teachers of Mathematics (NCTM) states that problem solving is an integral part of all mathematics learning and exposure to problem solving strategies should be embedded across the curriculum. Furthermore, by high school, students should be able to use, decide and invent a wide range of strategies.…

In an elementary undergraduate abstract algebra or group theory course, a student is introduced to a variety of methods for constructing and deconstructing groups. What seems to be missing from contemporary texts and syllabi is a theorem, first proved by Edouard Jean-Baptiste Goursat (1858-1936) in 1889, which completely describes the subgroups of…

The complexities involved in writing mathematics problems and a cooperative learning activity that gives students the challenge of writing mathematical logic problems for their peers is discussed. Making students construct mathematics problems taps into several important skills in mathematics and students are asked to consider what it means to…

This monograph was developed with the intention of addressing the concerns of high school mathematics teachers in Alberta (Canada) who want to base their programs on problem solving but have questions about effective and efficient ways to do so. Considered are the most basic philosophical questions, and a framework is provided to use in solving…

Describes a model for geometrical probability. Presents two examples of basic theories of probability using geometrical probability. Provides three problems using the described theorem. (YP)

Presents an example from the combinatorial theory of partially ordered sets. Discusses algorithms of on-line antichain partitioning problems, a topic in discrete optimization. (YP)

Describes investigations of quick cancellation problems in two-digital integers. Generalizes the results to n-digital integer problems. (YP)

Discusses how to express an n factorial as a product of powers of primes. Provides two examples and answers. Presents four related suggestions. (YP)

Provides several problems and their solutions for a descriptive statistics course. Included are problems related to measurement, data collection, sample construction, applications, and computation. Discusses the misleading nature of statistics. (YP)

Discusses geometric construction problems. Presents four ways to construct a regular octagon using different conditions. Provides drawings showing the constructions. (YP)