Challenging the traditional sequence of teaching introductory calculus

Despite considerable research with students of calculus, rate, and hence derivative, remain difficult concepts to teach and learn. The demonstrated lack of conceptual understanding of introductory calculus limits its usefulness in related areas. Since rate is such a troublesome concept, this study piloted reversing the usual presentation of introductory calculus to begin with area and integration, rather than rate and derivative. Two classes of first-year university students taking introductory calculus were selected to pilot the effect of changing the sequence; one class was a control group and the other class followed the reversed sequence. Advances in technology, especially computer algebra systems (CASs) may facilitate new ways of studying mathematics. In this study, handheld CASs were used to support students’ thinking as they grappled with the concepts of introductory calculus. The use of CASs enabled consideration of symbolic patterns and numerical integration leading to a deeper conceptual understanding of integration. The easy access to the multiple representations of functions provided by CASs facilitated an exploration of rate where each representation highlighted different aspects of rate resulting in deeper conceptual understanding of differentiation.

An approximate numerical technique for characterizing optical pulse propagation in inhomogeneous biological tissue

An approximate numerical technique for modeling optical pulse propagation through weakly scattering biological tissue is developed by solving the photon transport equation in biological tissue that includes varying refractive index and varying scattering/absorption coefficients. The proposed technique involves first tracing the ray paths defined by the refractive index profile of the medium by solving the eikonal equation using a Runge-Kutta integration algorithm. The photon transport equation is solved only along these ray paths, minimizing the overall computational burden of the resulting algorithm. The main advantage of the current algorithm is that it enables to discretise the pulse propagation space adaptively by taking optical depth into account. Therefore, computational efficiency can be increased without compromising the accuracy of the algorithm.

Numerical modeling and analysis for forming process of dual-phase 980 steel exposed to infrared local heating

A recent experiment confirmed that the infrared (IR) local heating method drastically reduces springback of dual-phase (DP) 980 sheets. In the experiment, only the plastic deformation zone of the sheets was locally heated using condensed IR heating. The heated sheets were then deformed by V-bending or 2D-draw bending. Although the experimental observation proved the merit of using the IR local heating to reduce springback, numerical modeling has not been reported. Numerical modeling has been required to predict springback and improve the understanding of the forming process. This paper presents a numerical modeling for V-bending and 2D-draw bending of DP 980 sheets exposed to the IR local heating with the finite element method (FEM). For describing the thermo-mechanical behavior of the DP 980 sheet, a flow stress model which includes a function of temperature and effective plastic strain was newly implemented into Euler-backward stress integration method. The numerical analysis shows that the IR local heating reduces the level of stress in the deformation zone, although it heats only the limited areas, and then it reduces the springback. The simulation also provides a support that the local heating method has an advantage of shape accuracy over the method to heat the material as a whole in V-bending. The simulated results of the springback in both V-bending and 2D-draw bending also show good predictions.