New Findings in Old Geometry: Using Triangle Centers to Create Similar or Congruent Triangles

The Euler line of a triangle passes through several important points, including three specific triangle centers: the centroid, orthocenter, and circumcenter. Each of these centers is the intersection of lines related to the triangle, mainly its medians, altitudes, and perpendicular bisectors, respectively. We present three theorems which initially share a similar construction. Each involves starting with a triangle and a point. After connecting the triangle’s vertices to that point, creating additional triangles, we establish connections to either the centroids, orthocenters, or circumcenters of the new triangles

Mathematics Versus Statistics

Mathematics and statistics are both important and useful subjects, but the former has maintained prominence in the American education system. On the other hand, statistics is more prevalent in daily life and is an increasingly marketable subject to know. This article gives a personal history of one mathematician’s bumpy road to learning and teaching statistics. Additionally, arguments for how and why to include statistics in the K-12 and college curricula are provided

Carcassonne in the Classroom

“What is the probability of choosing a green ball from an urn with three blue balls,five green balls, and seven yellow balls?” Many students not only struggle to engage with this sort of question but are left wondering why the world of mathematics is obsessed with balls and urns. The variety of approaches to choose from makes probability a difficult subject for many students, yet probability is an important part of quantitative literacy since it is prevalent in everyday life. Finding ways to clarify probability for undergraduates is key to a successful mathematics experience. One strategy for increasing student engagement with probability concepts is to teach probability through its application to games. Many previous works have investigated the use of Markov chains to model board games, such as Chutes and Ladders [2, 4, 5], Monopoly [1, 3], and Risk [6, 7, 8]. While these works have primarily focused on understanding the various games for their own sake, in this article we focus on using the board game Carcassonne in the classroom as a path for students to learn about probability through a more interesting context. In particular, we give a sequence of increasingly difficult probability problems derived from Carcassonne that can be used in a wide range of undergraduate mathematics courses

Investigating Prerequisite Grade Requirements in the Calculus Sequence

The Valparaiso University Mathematics and Computer Science Department has been debating a shift from a D- to a C- in the calculus sequence prerequisite grade requirement. We first researched policies at other colleges and universities. Next, we gathered data on students\u27 self-predicted grades by distributing pre- and post-semester surveys and measuring the accuracy of the results in comparison to the actual grade data. Using the surveys, we also associated students\u27 expectations with their majors. In the end, we investigated whether or not prerequisite grades can predict subsequent course performance. Statistical analysis of the past four years of math student data determined that a student who received a C or lower in the prerequisite class generally would not improve in the subsequent course

The Use of Exam Notes in an Online Mathematics Course

Many students want to use cheat sheets, or crib notes, on exams. Whether or not those aids actually help them has not been carefully studied. This paper measures 16 students’ notes by scoring the writing density as well as the number of definitions, examples, and mistakes. To consider the effectiveness of the notes, they are matched against exam solutions. This closer look at the content and application of the online students’ notes revealed that they were neither well- made nor well used

The Use of Exam Notes in an Online Mathematics Course

Many students want to use cheat sheets, or crib notes, on exams. Whether or not those aids actually help them has not been carefully studied. This paper measures 16 students’ notes by scoring the writing density as well as the number of definitions, examples, and mistakes. To consider the effectiveness of the notes, they are matched against exam solutions. This closer look at the content and application of the online students’ notes revealed that they were neither well- made nor well used

What Definitions are Your Students Learning?

Definitions are fundamental to any mathematics classroom. We consider how definitions are presented in textbooks and instructors’ lecture notes and whether students are positioned to gain a full understanding of new terms. Four textbooks, covering precalculus, calculus, and analysis, as well as three professors’ Calculus I lecture notes are compared. We found that textbooks often include two versions of definitions, formal and informal, but those definitions differ across texts. Instructors’ lecture notes do not usually have more than one version of a definition, and that version may not align with the course textbook, but they do sometimes offer intuitive or visual interpretations. We provide suggestions for being more mindful when deciding how and what definitions to present

Inquiry-Based Learning in Mathematics

Most college students are required to take at least one mathematics course. Many of these students view mathematics as a dry and tedious subject, where the main task is to “plug and chug” using formulas. In contrast, mathematicians see mathematics as a creative process in which real joy comes from grappling with difficult problems and (hopefully) solving them. In this way, mathematics is like a fun puzzle. The challenge is to get students to view mathematics the same way that their teachers do. Inquiry-based learning (IBL) can help solve this problem. The Academy of Inquiry-Based Learning describes IBL as a pedagogical method that encourages students to conjecture, discover, solve, explore, collaborate, and communicate (What is IBL? (n.d.). Retrieved from http://www.inquirybasedlearning.org/?page=What_is_IBL). With IBL, teachers do not lay out all of the formulas and theorems as previous knowledge. Nor do they provide perfect, easily worked through examples and proofs for every new topic. Instead, IBL courses demonstrate the creative process that is mathematics. IBL makes class more enjoyable for both teachers and students, and can bring students closer to the real experiences of mathematicians

Including Inquiry-Based Learning in a Flipped Class

Flipped classrooms and inquiry-based learning (IBL) have each become popular in their own right, leading to a natural question: Why not combine these two great ideas? Although flipping a class usually involves students reading or watching videos before class, and IBL focuses on allowing and encouraging students to develop material on their own, both styles emphasize active learning and critical thinking through activities such as group work and presentations while minimizing lectures. In this article, I discuss ways that the two teaching styles can complement each other and be implemented concurrently, with some examples from my flipped calculus II course. Throughout this discussion the focus remains on ways to keep students engaged and how to instill deep content knowledge