As new technological developments continue to change the educational landscape, it is not an exception in the area of proof and proving. This classroom note introduces the use of one of the trending proofs assistants -- the Lean theorem prover. We first provide a technical account of Lean, then exemplify Lean proofs in propositional logic, number…

Bell's theorem is a topic of perennial fascination. Publishers and the general public have a steady appetite for approachable books about its implications. The scholarly literature includes many analogies to Bell's theorem and simple derivations of Bell inequalities, and some of these simplified discussions are the basis of interactive web pages.…

The mathematical egg hunt is a hands-on activity designed to help students understand mathematical relations in an Introduction to Proofs course. This activity gives students the opportunity to practice selecting which ordered pairs do and do not belong to a given relation in a moderately competitive egg hunt. It is designed to be low-stakes, yet…

We present an investigation of the infinite sequences of numbers formed by calculating the pairwise averages of three given numbers. The problem has an interesting geometric interpretation related to the sequence of triangles with equal perimeters which tend to an equilateral triangle. Investigative activities of the problem are carried out in…

While noncompliance is a concerning challenging behavior and commonly reported by educators, its measurement is likely to be invalid and inaccurate given the subjectivity of the operational definition. Engagement is offered as a more valid, accurate measurement that may provide data regarding the amount of instruction accessed by the student. In…

LaPiere's pioneering study explored the tenuous connection between attitudes and behaviors in the context of racism. This activity provides a replication proposal framework that is useful for applying critical thinking skills, including new forms of measurement. Students read the study and provide written responses to several discussion prompts…

This paper presents a novel figure for teaching multiple geometric proofs of the Pythagorean theorem. Because it consists only of congruent given right triangles, the figure can be constructed using a template of the given right triangle or, if available, a computer program. Within the figure, called a Pythagorean multi-proof square, there are…

Knowledge assessments in undergraduate mathematics education commonly evaluate response correctness to determine learner proficiency. However, simultaneous evaluation of learner metacognition more accurately assesses the multiple dimensions of knowledge and has been shown to increase assessment validity and reliability. Research into…

This article aims to share five practical approaches for how teachers can identify, adapt, and create their own writing prompts they assign to students: (1) Promote Students' Solution Paths; (2) Go beyond Asking Students to Simply "Explain"; (3) Prompt Students to Share Their Reasoning; (4) Have Students Consider the Validity of a Given…

A simple in-class demonstration of integral Calculus for first-time students is described for straightforward whole number area magnitudes, for ease of understanding. Following the Second Fundamental Theorem of the Calculus, macroscopic differences in ordinal values of several integrals, [delta]"F"(x), are compared to the regions of area…

The purpose of this resource is to help math teachers unpack, understand, and implement the current math content and practice standards. This resource describes the progressions of learning within each course and provides content support, which includes broad ideas about effective instruction as well as practical instructional strategies. Math…

At the highest level, this dissertation is a case study on how bias can become encoded into the tools used to measure a construct and into the very definition of the construct itself. In this case, the construct is spatial ability. This dissertation focuses on the validity and accuracy of spatial tests and illuminates gender bias that is…

The author discusses valid and reliable ways to assign grades in an academic course in any discipline. "Validity" means the accuracy of a grade's reflection of student learning and achievement. "Reliability" concerns the degree of random error that might be present and affect validity. First the author defines a grade as a…

This book addresses the validity of think-aloud protocols (TAPs) in L2 writing research through a mixed methods study and proposes effective approaches for their valid implementation. The book uncovers the reactive effects that TAPs have on L2 writing performance and processes, and examines how individual factors moderate this reactivity. It…

We have revamped a classic analytical chemistry laboratory experiment, "Determination of an Unknown Acid", to provide students an opportunity to learn about new advances in the material sciences and their applications in analytical chemistry in an effort to reinforce the key concepts of chemical analysis. Specifically, students were…