The author introduces the concept of the quantum volume into thermodynamics teaching. A new derivation of the basic part of the Sackur-Tetrode equation uses the microcanonical ensemble instead of the math-heavy canonical ensemble. Entropy diagrams, based on the binomial distribution, visualize the ratio between the quantum volume and system volume.

The goal of this study is to show a novel way of using large-scale data (N = 6203) to identify pupils' strategies when solving missing value number equations. It is based on the assumption that wrong numerical results appearing more frequently than would be the case if they were consequences of random guessing can be expected to be underlain by a…

Research over the past 20 years has suggested that our intuitive sense of number--the Approximate Number System (ANS)--is associated with individual differences in symbolic math performance. The mechanism supporting this relationship, however, remains unknown. Here, we test whether the ANS contributes to how well adult observers judge the…

We use Taylor's formula with Lagrange remainder to prove that functions with bounded second derivative are rectifiable in the case when polygonal paths are defined by interval subdivisions which are equally spaced. As a means for generating interesting examples of exact arc length calculations in calculus courses, we recall two large classes of…

The key difference between math as math and math in science is that in science we blend our physical knowledge with our knowledge of math. This blending changes the way we put meaning to math and even the way we interpret mathematical equations. Learning to think about physics with math instead of just calculating involves a number of general…

In this paper I explain how I usually introduce the Schrödinger equation during the quantum mechanics course. My preferred method is the chronological one. Since the Schrödinger equation belongs to a special case of wave equations I start the course with introducing the wave equation. The Schrödinger equation is derived with the help of the two…

Many people find that solving story problems is challenging. A key source of the difficulty is symbolizing problem situations. In two experiments, we tested whether diagrams promoted successful symbolization of two-operator story problems, and we tested whether diagrams that depicted the integration of operations were more beneficial than diagrams…

Bernstein polynomial is one of the most valuable and attractive method used to develop numerical solution for several complex models because of its robustness to demonstrate approximation for anonymous equations. In this paper, Bernstein polynomial is proposed to present effective solution for the 2nd kind linear Volterra integral equations with…

In this chapter we are interested in how undergraduate physics students in three countries experience the equations they meet in their education. We asked over 350 students in the USA, Australia and Sweden the same simple question: How do you know when you understand a physics equation? Students wrote free-text answers to this question, and these…

Laplace transform (LT) is an essential mathematical tool for solving linear ordinary differential equations (ODE) with boundary values, by transforming differential equation into algebraic equations which are easier to manipulate. In this article, we analyse the errors students make and misconceptions they have in solving linear ODE using LT…

In this study, we derived several formulas for the currents induced in a circular loop by a magnet connected to a spring-based simple harmonic oscillation system. In addition, we conducted an experiment for measuring the induced currents and compared the results with the theoretical prediction. It was confirmed that the prediction from the derived…

In this study we aimed to inquire into the impact of the use of early arithmetic strategies by a group of fifth-grade students, on their solving of equations involving two representations of unknowns. Pre- and post-tests consisting of equation-solving items involving two representations of unknowns (number sentences containing empty 'brackets', as…

Quadratic equations are a notorious topic for the challenge it provides to students in secondary mathematics. Despite this, there is limited research, particularly in the Australian context, that explains why such challenges persist. This article details the causes of Year 11 students' difficulties in solving quadratic equations. Observing…

Increasingly individuals with Intellectual Disability (ID) are showing the capability of learning abstract mathematical skills like algebra. The purpose of the study was to show a method for teaching high school-aged individuals with ID the algebra skill of creating an equation from a line using a time-delay strategy and equation template. We…

The aim of this paper is to identify errors and misconceptions that student demonstrated when learning linear independence and linear dependence concepts. A case study is presented involving 73 in-service mathematics teachers at a university in Zimbabwe who were studying for a Bachelor of Science Education Honors Degree in mathematics. Data was…