Knowing an equation has a unique solution is important from both a modelling and theoretical point of view. For over 70 years, the approach to learning and teaching "well posedness" of initial value problems (IVPs) for second- and higher-order ordinary differential equations has involved transforming the problem and its analysis to a…

In this article we present the results of an investigation into semiotic aspects of an introductory differential equations class from two countries. In order to perform a comparative analysis, students in the USA and Tunisia were given common questions on their midterm exams that examined analytical and graphical competencies. Our results indicate…

Modeling projects in differential equations and engineering courses create an authentic activity and an opportunity to attain the holy grail of "deeper" learning. However, what do we mean by "deeper" learning and how do we create an environment that encourages that? This article describes a proposal and case study in using…

For over 50 years, the learning of teaching of "a priori" bounds on solutions to linear differential equations has involved a Euclidean approach to measuring the size of a solution. While the Euclidean approach to "a priori" bounds on solutions is somewhat manageable in the learning and teaching of the proofs involving…

Recently we were presented with an interesting twist to the sliding ladder problem viewed in the related rates section of most calculus textbooks. Our problem concerning a sliding ladder that eventually hits the ground. At first, those attempting this problem fell into the calculus trap using only related rates. Previous work for this problem…

Over the last number of years we have gradually been introducing a project based learning approach to the teaching of engineering mathematics in Dublin Institute of Technology. Several projects are now in existence for the teaching of both second-order differential equations and first order differential equations. We intend to incrementally extend…

Bayesian structural equation modeling (BSEM) was used to investigate the latent structure of the Differential Ability Scales-Second Edition core battery using the standardization sample normative data for ages 7-17. Results revealed plausibility of a three-factor model, consistent with publisher theory, expressed as either a higher-order (HO) or a…

The authors propose an integrative theoretical model of reading called the direct and indirect effects model of reading (DIER) that builds on and extends several prominent theoretical models of reading. According to DIER, the following skills and knowledge are involved in reading comprehension: word reading, listening comprehension, text reading…

The authors propose an integrative theoretical model of reading called the direct and indirect effects model of reading (DIER) that builds on and extends several prominent theoretical models of reading. According to DIER, the following skills and knowledge are involved in reading comprehension: word reading, listening comprehension, text reading…

We investigated 2 hypotheses of a recently proposed integrative theoretical model of reading, the direct and indirect effects model of reading (DIER; Kim, 2017b, 2019): (a) hierarchical relations and (b) dynamic relations (or differential relations) of skills to reading comprehension. Students were assessed on reading comprehension, word reading,…

We give an alternative to the standard method of reduction or order, in which one uses one solution of a homogeneous, linear, second order differential equation to find a second, linearly independent solution. Our method, based on Abel's Theorem, is shorter, less complex and extends to higher order equations.

We investigated 2 hypotheses of a recently proposed integrative theoretical model of reading, the direct and indirect effects model of reading (DIER; Kim, 2017b, 2019): (a) hierarchical relations and (b) dynamic relations (or differential relations) of skills to reading comprehension. Students were assessed on reading comprehension, word reading,…

Drawing on expectancy-value theory, the present study examined the unique contributions of the four major value beliefs and self-concept on achievement, self-reported effort, and teacher-rated behavioral engagement in mathematics. In particular, we examined the multiplicative effects of self-concept and task values on educational outcomes using…

Are there functions for which Newton's method cycles for all non-trivial initial guesses? We construct and solve a differential equation whose solution is a real-valued function that two-cycles under Newton iteration. Higher-order cycles of Newton's method iterates are explored in the complex plane using complex powers of "x." We find a class of…

Academic self-concept has been conceptualized as a multidimensional and hierarchical construct. Previous research has mostly focused on its multidimensionality, distinguishing between verbal and mathematical self-concept domains, and only a few studies have examined the factorial structure within specific self-concept domains. The present study…