This article presents four inquiry-based learning activities developed for a liberal arts math course. The activities cover four topics: the Pythagorean theorem, interest theory, optimization, and the Monty Hall problem. Each activity consists of a dialogue, with a theme and characters related to the topic, and a manipulative, that allow students…

The goal of the study was to reconstruct and dismantle a sequence of events that preceded an insight solution to a challenging problem by a ninth-grade student. A three-week long solution process was analysed by means of the theory of shifts of attention. We argue that concurrent focusing on what, how and why the student attends to when working on…

In this note, we introduce sequence factorial and use this to study generalized M-bonomial coefficients. For the sequence of natural numbers, the twin concepts of sequence factorial and generalized M-bonomial coefficients, respectively, extend the corresponding concepts of factorial of an integer and binomial coefficients. Some latent properties…

We present a sequence of improper integrals, for which a closed formula can be computed using Wallis formula and a non-straightforward recurrence formula. This yields a new integral presentation for Catalan numbers.

A new application of logarithmic differentiation is presented, which provides an alternative elegant proof of two basic rules of differentiation: the product rule and the quotient rule. The proof can intrigue students, help promote their critical thinking and rigorous reasoning and deepen their understanding of previously encountered concepts. The…

We consider an oblique approach to cutting regions out of a flat rectangular sheet and folding to make a maximum volume container. We compare our approach to the traditional approach of cutting out squares at each vertex of the sheet. (Contains 4 figures.)

For a given positive integer k [not equal] 4, let "P[subscript k,n]" denote the "n"-th "k"-gonal number. We study "k"-gonal triples ("a", "b", "c") satisfying P[subscript k,a] + P[subscript k,b] = P[subscript k,c]. A "k"-gonal triple corresponds to a rational point on the rectangular hyperboloid x[squared] + y[squared] = z[squared] + 1. The simple…

Permutations and combinations are used to solve certain kinds of counting problems, but many students have trouble distinguishing which of these concepts applies to a given problem. An "order heuristic" is usually used to distinguish the two, but this heuristic can cause confusion when problems do not explicitly mention order. This…

This paper presents views on the teaching of chemistry and directions for its further development. A detailed critical analysis is offered for the inadequacy of much of the current teaching, weighed that it is by a conventional, traditional and, as it turns out, rather outdated sense of the material to be covered. The ambient meta-discourse on the…

In the author's eighth year of teaching, she hit a wall teaching percent change. Percent change is one of the few calculations taught in math classes that shows up regularly in the media, and one that she often does in her head to make sense of the world around her. Despite this, she had been teaching percent change using textbook problems about…

This paper presents a simplified approach for the application of material efficiency metrics to linear and convergent synthesis plans encountered in organic synthesis courses. Computations are facilitated and automated using intuitively designed Microsoft Excel spreadsheets without invoking abstract mathematical formulas. The merits of this…

This study investigated how three factors impacted performance on cost-volume-profit homework problems: language, formula use, and instruction. Students enrolled in Introduction to Financial Accounting (the first principles of accounting course) and Managerial Accounting (the second principles of accounting course) from eight different US colleges…

Early in their chemistry education, students learn to do empirical formula calculations by rote without an understanding of the historical context behind them or the reason why their calculations work. In these activities, students use paperclip "atoms", construct a series of simple compounds representing real molecules, and discover,…

Shulman's notations of subject matter knowledge (SMK) and pedagogical content knowledge (PCK) have been very influential in education research on teachers' knowledge for teaching. However, there is little empirical evidence in support of these as separate analytical constructs. Furthermore, attempts to distinguish SMK and PCK highlight the complex…

What does "to be a mathematician" mean? What is implied, and what image is created of "a mathematician"? Are "mathematicians" members of an exclusive club? Are mathematicians different to "other people"? Are mathematicians different because they are able to mathematize? These might not be the most oft asked questions, but are they questions to…