The emergence of non-Euclidean geometries in the 19th century rocked the foundations of mathematical knowledge and certainty. The tremors can still be felt in undergraduate mathematics today where encounters with non-Euclidean geometry are novel and often shocking to students. Because of its divergence from ordinary and comfortable notions of…

Identifying mathematical and didactical conditions under which mathematics learners can encounter an intellectual need for defining and proving is recognized as a challenging research enterprise. This paper presents a particular configuration of conditions under which a group of pre-service mathematics teachers successfully constructed a…

This article investigates the numbers [image omitted], originally studied by Catalan. We re-confirm that they are indeed integers. Using the close relationship between them and the Catalan numbers C[subscript n], we develop some divisibility properties for C[subscript n]. In particular, we establish that [image omitted], where f[subscript k]…

This paper shows how to teach a mathematics for liberal arts class in an inquiry-based way using ideas from music to launch the mathematical activities. No musical knowledge is required to understand and teach the material. The main activity is analyzing the differences between two kinds of rhythmic palindromes. The content is mathematically…

The late-Victorian era provided aficionados of detective fiction with an abundance of short stories belonging to this literary genre. Many of these works contained some aspect of chemistry, either in the execution of a crime or in the identification of the perpetrator. In the example discussed in this article, the gas-phase reaction of hydrogen…

This paper presents guided classroom activities that showcase two classic problems in which a finite limit exists and where there is a certain charm to engage liberal arts majors. The two scenarios build solely on students' existing knowledge of number systems and harness potential misconceptions about limits and infinity to guide their thinking.…

This article summarizes research conducted on calculator block items from the 2007 fourth- and eighth-grade National Assessment of Educational Progress Main Mathematics. Calculator items from the assessment were categorized into two categories: problem-solving items and noncomputational mathematics concept items. A calculator has the potential to…

Dewey's conception of scientific explanation, which has been neglected by both philosophers of science and philosophers of education, facilitates overcoming the seeming divide between teaching a highly technical and specialized subject matter and encouraging students to successfully engage in the experience of being philosopher-scientists. By…

This article describes a two-day lesson in which fifth-grade students took on the role of marine biology scientists, using their critical-thinking and problem-solving skills to explore human impact on the blue crab ecosystem. The purpose of "Crabby Interactions" was to help students understand the impact of human activities on the local…

In this first of a two-part series, the authors report that STEM is still mostly science and mathematics, taught separately with little or no attention to technology and engineering. Where connections do get made to technology and engineering, too often they happen through several disconnected projects that lack coherence in content standards and…

The Year Abroad (YA) is arguably the most valued feature of any languages degree, according to employers (British Academy & UCML, 2012). The resilience and problem-solving capacities developed by students who undertake a period of residence abroad are sought-after characteristics in the job market. Thus, most languages degrees in the UK…

This paper describes how the author's students (in-service and pre-service secondary mathematics teachers) enrolled in college geometry courses use the Geometers' Sketchpad (GSP) to gain insight to formulate, confirm, test, and refine conjectures to solve the classical airport problem for triangles. The students are then provided with strategic…

Within the "Australian Curriculum: Mathematics" the Understanding proficiency strand states, "Students build understanding when they connect related ideas, when they represent concepts in different ways, when they identify commonalities and differences between aspects of content, when they describe their thinking mathematically and…

It was a hot summer day. Naomi and her cousin walked into the community center. She said, "Miss Jones, our moms need to work. They told us to come here and stay for couple hours." This is a common occurrence in communities with low socioeconomic status during the summertime; parents need to go to work, but children are on summer break.…

Sometimes a student's unexpected solution turns a routine classroom task into a real problem, one that the teacher cannot resolve right away. Although not knowing the answer can be uncomfortable for a teacher, these moments of uncertainty are also an opportunity to model authentic problem solving. This article describes such a moment in Zahner's…