The "Unsolved Problems in Number Theory" book by Richard Guy provides nice problems suitable for a typical math major. We give examples of problems that have worked well in our senior seminar course and some nice results that senior math majors can obtain.

In this paper I discuss the motivation behind using an introduction to Fourier series as a method to unify the topics of integration techniques and series calculations in a Calculus II course.

This article outlines a problem-centered approach to the topic of canonical matrix forms in a second linear algebra course. In this approach, abstract theory, including such topics as eigenvalues, generalized eigenspaces, invariant subspaces, independent subspaces, nilpotency, and cyclic spaces, is developed in response to the patterns discovered…

Multi-voxel pattern recognition techniques combined with Hidden Markov models can be used to discover the mental states that people go through in performing a task. The combined method identifies both the mental states and how their durations vary with experimental conditions. We apply this method to a task where participants solve novel…

With crosscutting concepts such as stability and change in the "Next Generation Science Standards," this article was written for those who have wondered how to teach these concepts in a way that is relevant to students. In this investigation, students ask the question, "Why is the pond dirty?" As students investigate the health…

Having a deeper understanding of division derived from multiple models is of great importance for teachers and students. For example, students will benefit from a greater understanding of division contexts as they study long division, fractions, and division of fractions. The purpose of this article is to build on teachers' and students'…

In this article, the author considers the merits of two classes of profit maximization problems: those involving perfectly competitive firms with quadratic and cubic cost functions. While relatively easy to develop and solve, problems based on quadratic cost functions are too simple to address a number of important issues, such as the use of…

In 1879, in the midst of the debate between English and continental scientists about the nature of cathode rays, William Crookes conducted an experiment in which a small mill or "paddle wheel" was pushed along tracks inside a cathode ray tube (CRT) (similar to that shown in Fig. 1) when connected to a high-voltage induction coil. Crookes…

I am a volunteer reader at the Princeton unit of "Learning Ally" (formerly "Recording for the Blind & Dyslexic") and I recently discovered that high school students are introduced to the concept of quantization well before they take chemistry and physics. For the past few months I have been reading onto computer files a…

This module examines Rigor, the third of the three shifts required for implementation of the Common Core State Standards for Mathematics (CCSSM). This module is based on the premise that the information from Modules 1 and 2 is well understood. Course Objectives: By the end of the module, the learner will be able to: (1) identify the components of…

The goal of the present article is to provide the teacher with an alternative perspective in writing multiple-choice questions. To the techniques and advice available in the literature on the topic, three further aspects are added whose provenance is an emerging line of research--the exemplification of concepts. The objective is to expand…

The article investigates the patterns that arise in the convergence of numerical methods, particularly those in the errors involved in successive iterations, using data analysis and curve fitting methods. In particular, the results obtained are used to convey a deeper level of understanding of the concepts of linear, quadratic, and cubic…

Motion planning is a core problem in robotics concerned with finding feasible paths for a given robot. Motion planning algorithms perform a search in the high-dimensional continuous space of robot configurations and exemplify many of the core algorithmic concepts of search algorithms and associated data structures. Motion planning algorithms can…

Our study concerns the conceptual mathematical knowledge that emerges during the resolution of tasks on the equivalence of polynomial and rational algebraic expressions, by using CAS and paper-and-pencil techniques. The theoretical framework we adopt is the Anthropological Theory of Didactics ("Chevallard" 19:221-266, 1999), in…

This paper deals with the nature and the level of difficulty of teaching and learning physics in the first year of undergraduate engineering schools in France. Our case study is based on a survey regarding a classic and basic question in applied physics, and which was conducted with a group of second-year students in a post-baccalaureate 1…