As educational systems move from onsite to online due to the COVID-19 pandemic, teachers face the difficulty of designing online examination formats which minimize opportunities for dishonesty. In this paper, we expose our design of such a format: a protected Microsoft Excel spreadsheet containing short-answer questions, which was implemented in a…

Virtual learning environments give students more autonomy over their learning than traditional face-to-face classes and require that students adapt the ways they consume and assimilate new information. One theory of this process is self-regulated learning, which is illustrated in Efklides' Metacognitive and Affective model of Self-Regulated…

The motion of a bead along a path restricted to straight lines (restricted brachistochrone), sliding without friction from rest and accelerated by gravity, is considered. For two shapes of path, the geometry of the route optimized to provide the least travel time from one point to another is obtained. The bead's travel times, path lengths, and…

Differential and integral calculus textbooks are widely used as the main resource for teaching. They appear in a variety of forms and adopt various approaches to present the content. In this paper, we turn our attention to one chapter of a calculus textbook and our focus is on the introduction of the derivative concept. With the purpose of…

For a function "f": [real numbers set][superscript n]\{(0,…,0)}[right arrow][real numbers set] with continuous first partial derivatives, a theorem of Euler characterizes when "f" is a homogeneous function. This note determines whether the conclusion of Euler's theorem holds if the smoothness of "f" is not assumed. An…

The present study aimed to identify the errors made by pre-service elementary mathematics teachers while investigating the convergence of infinite series. A qualitative exploratory case study design was used with a total of 43 undergraduate students. Data were obtained from a test administered in a paper-and-pencil form consisting of seven…

I describe the motivation, structure, implementation, and student feedback of my mastery-based testing (MBT) system with a significant final exam component in Calculus I and II since 2016. Grades for each written assignment, quiz, and exam question were assigned on a Mastery/Progressing/Needs Improvement (M/P/NI) scale. Homework and quizzes could…

Acknowledging the contribution of mathematicians to the mathematical education of teachers, we explore mathematicians' perspective on an envisioned Calculus course for prospective teachers. We analyzed semi-structured interviews with 24 mathematicians using the EDW (Essence-Doing-Worth) framework (Hoffmann & Even, 2018, 2019); and…

We use Taylor's formula with Lagrange remainder to make a modern adaptation of Poisson's proof of a version of the fundamental theorem of calculus in the case when the integral is defined by Euler sums, that is Riemann sums with left endpoints which are equally spaced. We discuss potential benefits for such an approach in basic calculus courses.

In our modern world, we are inundated and grapple with data daily. As mathematicians, we are often more comfortable discussing the behavior of functions presented analytically, in contrast with the data-driven or tabular presentations of functions ubiquitous in our culture. This paper introduces an entry-level course, Mathematical Modeling and…

This paper demonstrates how the transcendental number "e" may be arrived at by observing the discharge of a capacitor through a fixed resistor and then modelling the system using a simple step-wise procedure. The experimental phase makes use of the Arduino microcontroller, while simple modelling of the system is carried out by means of…

Retrieving information from memory increases the likelihood the information will be remembered later. The strategic use of retrieval to enhance memory is known as retrieval practice. Teachers can exert considerable control over students' retrieval practice, dictating when and how much students practice. Laboratory research has shown that retention…

Effective mathematics teaching elicits and uses evidence of student thinking to assess progress toward mathematical understanding and adjusts instruction continually to support and extend learning (National Council of Teachers of Mathematics, 2014). However, as teachers march through content in precalculus, they tend to rely heavily on traditional…

This research paper explores the integration of ChatGPT as a tool for interactive learning of numerical methods in a differential equations (DEs) course. DE course is crucial for engineering students to model real-world phenomena. However, many DE courses focus only on analytical solutions and neglect important numerical solutions. To overcome…

Nationwide, many students fail to complete the key mathematics courses that are required for most STEM majors, including Precalculus. This paper describes the rationale, implementation, and impact of the redesign of Precalculus at one regional west coast institution. Prior to the redesign, pass rates in Precalculus were modest (75.8% of all…