A typical introductory course in ordinary differential equations (ODEs) exposes students to exact solution methods. However, many differential equations must be approximated with numerical methods. Textbooks commonly include explicit methods such as Euler's and Improved Euler's. Implicit methods are typically introduced in more advanced courses…

Proofs that the area of a circle is nr[superscript 2] can be found in mathematical literature dating as far back as the time of the Greeks. The early proofs, e.g. Archimedes, involved dividing the circle into wedges and then fitting the wedges together in a way to approximate a rectangle. Later more sophisticated proofs relied on arguments…

Several mathematicians struggled to solve cubic equations, and in 1515 Scipione del Ferro reportedly solved the cubic while participating in a local mathematical contest, but did not bother to publish his method. Then it was Cardano (1539) who first published the solution to the general cubic equation in his book "The Great Art, or, The Rules of…

Team-based learning (TBL) is a form of student-centered active learning in which students independently study new conceptual material before it is treated in the classroom, and then subsequently spend considerable classroom time working in groups on increasingly challenging problems and applications based on that new material. TBL provides…

The Common Core State Standards for Mathematics (CCSSM) (CCSSI 2010) identifies the strategic use of appropriate tools as one of the mathematical practices and emphasizes the use of pictures and diagrams as reasoning tools. Starting with the early elementary grades, CCSSM discusses students' solving of problems "by drawing." In later…

Problems with handwriting can negatively impact the writing of students with learning disabilities. In this article, an example is provided of a fourth-grade special education teacher's efforts to assist a new student by using a problem-solving approach to help determine an efficient course of action for special education teachers who are trying…

The need to address the substantial inequities that exist between Indigenous and non-Indigenous Australians in higher education is widely recognised. Those factors that affect the performance of Indigenous students in tertiary education have been reasonably well documented across different institutions, disciplines, and programme levels but there…

The achievement of deep learning by secondary students requires teaching approaches that draw students into task commitment, integrated curricula, and analytical thinking. By using real-world data sets in project based instructional units, teachers can guide students in analyzing, interpreting, and reporting quantitative data. Working with…

The author's teaching goal is always to make certain that his students, children and adults alike, have some reason for joining him on his mathematical journeys. The lesson he describes in this article provides him with the unusual opportunity to call attention to strategies that have had a long history and remarkable results. With careful…

As part of the author's doctoral program at the University of Hawai'i at Manoa, she conducted a study to determine whether and how former sixth-grade students at Waikiki School were using the thinking processes and strategies they were taught in elementary school when they were in middle school. In order to find this out, she interviewed eight…

Fermat's principle is considered as a unifying concept. It is usually presented erroneously as a "least time principle". In this paper we present some software that shows cases of maxima and minima and the application of Fermat's principle to the problem of focusing in lenses. (Contains 12 figures.)

A facility with signed numbers forms the basis for effective problem solving throughout developmental mathematics. Most developmental mathematics textbooks explain signed number operations using absolute value, a method that involves considering the problem in several cases (same sign, opposite sign), and in the case of subtraction, rewriting the…

An approximate formula for the period of pendulum motion beyond the small amplitude regime is obtained based on physical arguments. Two different schemes of different accuracy are developed: in the first less accurate scheme, emphasis is given on the non-quadratic form of the potential in connection to isochronism, and a specific form of a generic…

Understanding the functional neuroanatomy of planning and problem solving may substantially benefit from better insight into the chronology of the cognitive processes involved. Based on the assumption that regularities in cognitive processing are reflected in overtly observable eye-movement patterns, here we recorded eye movements while…

Five high-school chemistry teachers were asked to enact a lesson in which they posed a problem for which students were likely to generate solutions based on reasoning that was not aligned with accepted principles of chemistry. Four teachers selected a problem related to the stoichiometry of a reaction; the fifth chose a problem associated with…