Zhu Ying

The purpose of this paper is to report a part of a calculus research project, about the performance of a group of pre-service mathematics teachers on two tasks on limit and differentiation of the trigonometric sine function in which the unit of angle measurement was in degrees. Most of the pre-service teachers were not cognizant of the unit of angle measurement in the typical differentiation formula, and a number of participants recognized the condition on the unit of angle measurement but did not translate this to the correct procedure for performing differentiation. The result also shows that most of the participants were not able to associate the derivative formula with the process of deriving it from the first principle. Consequently, they did not associate it with finding . In the process of evaluating this limit, the pre-service teachers exhibited further misconceptions about division of a number by zero.

Black cumin seed oil (BCSO) contains a large number of bioactive compounds and thus has many medicinal health benefits and uses. Its high economic profits in the market lead to the frequent occurrence of adulterating this oil with cheaper edible oils such as grape seed oil, walnut oil. It is difficult to detect adulteration as the oil adulterant has similar physical characteristic and even similar chemical composition to the authentic oil. The development of an accurate and rapid analytical method using attenuated total reflection Fourier transform infrared (ATR-FTIR) spectroscopy is of essential importance for determination of authenticity of BCSO and quantification of oil adulterants. In this study, BCSO and grape seed oil (GSO) were mixed in various ratios to mimic the adulteration. A clustering group lasso method was developed by incorporating both the high correlation structure of spectral variables and the underlying group features into the model. Instead of assuming that groups are known a priori as does ordinary group lasso, the clustering group lasso infers groups of spectral features from the data and encourages spectral variables within a group to have a shared association with the response. The model using ATR-FTIR spectroscopy proved to be a powerful tool to quantify BCSO adulteration with high accuracy and can accurately predict the quantity of adulterant at levels as low as 5%. With a substantial reduction in number of spectral features, the clustering group lasso model shows a simple regression coefficient profile with improved interpretability as compared to the ordinary group lasso model and other penalized models. The spectral regions automatically selected for quantification of BCSO adulteration can be helpful for the interpretation of the major chemical constituents of BCSO regarding its anti-cancer and anti-inflammatory effects from a chemometric perspective.

Constructivism was first popularised by Bruner (1960). The underlying theme in Bruner’s theoretical framework is that learning is an active process in which learners construct new ideas or concepts based upon their prior knowledge. This chapter describes how constructivism can be realised in instruction through a lesson design involving a carefully crafted task on the topic of Gradient of Function Curves at a point. The task affords opportunities to activate and differentiate students’ prior knowledge to generate, explore, critique and refine methods for problem solving. The lesson design allows teachers to first understand what students know about a new concept based on students’ representation and solution methods (RSMs) collected from the group work before the teacher teaches the canonical concept during lesson consolidation. The task, coupled with skillful facilitation and lesson consolidation built upon students’ RSMs, can help students develop a deep understanding of the targeted concept. Implications of such constructivist learning design on teachers’ classroom practice are also discussed.

Wild-grown Ganoderma lucidum (G. lucidum), a traditional Chinese herbal medicine, is highly cherished and expensive for its medicinal efficiency. This study targets the development of an accurate and effective analytical method to distinguish wild-grown G. lucidum from cultivated ones, which are of essential importance for the quality assurance and estimation of its medicinal value. Furthermore, different parts of G. lucidum have been studied to examine the differences between wild-grown and cultivated ones. Fourier transform infrared (FTIR) diffuse reflectance spectroscopy combined with the appropriate chemometric method has been proven to be a rapid and powerful tool for discrimination of wild-grown and cultivated G. lucidum with classification accuracy of 98%. The informative spectral absorption bands for discrimination emphasized by the linear diagnostic rule have provided quantitative interpretations of the chemical constituents of wildgrown G. lucidum regarding its anticancer effects.

This paper reports an exploratory study on the pre-service teachers’ content knowledge on school calculus. A calculus instrument assessing the pre-service teachers’ iconic thinking, algorithmic thinking and formal thinking related to various concepts in school calculus was administered to a group of pre-service mathematics teachers. Their performance on five of the items is reported in this paper. Other than their good performance in the iconic recognition of stationary points, their recognition on points of inflexion, differentiability and notion of minimum points was relatively poor. In addition, they appeared to lack the algorithmic flexibility in testing the nature of stationary points and the formal thinking about definition of an extremum point. The implications of the findings are discussed.