Study of instantaneous rate of change in a historical context

This study is essentially comprised of four parts. The primary purpose of the first part is to analyze the historical development of the concept of instantaneous rate of change, determine, in particular, the reasons for its comparatively late induction into the realm of natural sciences, and explore the uniquely significant role it has played in transforming mathematics as well as physics into their modern forms. The second part is devoted to the development of various learning models as applied to mathematics. In the third part we will analyze certain issues concerning foundations of mathematics and their impact on mathematics education. Finally, we will expound on how a well-designed array composed of the historic development of the notion of instantaneous rate of change, a suitable pedagogical model, and a correct philosophical foundational approach would elucidate certain fundamental concepts in geometry, algebra, and calculus; enrich students' understanding and appreciation of mathematics; and alter the common misperception that mathematics is merely a list of facts, by engendering a viable alternative to the usual prosaic teaching styles associated with these topics. We will also investigate possible associations between students' conception of mathematics and their academic background, academic affiliations, and future career choices. To collect the data required to establish our contention in as proficient a manner as possible, our research will be conducted in several parts. First, we will start out by an extensive questionnaire. Both our survey group (the experiment group) and our control group will be composed of students selected at random from a population of students taking precalculus and calculus classes at some local universities and colleges as well as at one overseas university. Secondly, with those members of the survey group who agree to participate, face-to-face interviews will be conducted after the completion of their next level mathematics course, and the extent to which the techniques they acquired while working on these projects contributed to the depth of their mathematical understanding and carried over to the next level will be measured. As a third step, in order to gauge students' progressive developments, a group of students (different than the original survey and control groups) selected from the same population will be surveyed on their attitudes towards mathematics at the beginning of the semester and then at the end of the semester (same questions) after they have been exposed to history of mathematics and to research projects involving history of mathematics. Lastly, to get some input from the teachers whose classes were involved in this research, they will be asked to provide us with some information on to what extend they covered history of mathematics in their classes prior to assigning projects, what sources they used, what handouts they distributed, and what were some of their project topics. We will then proceed to make some recommendations that, in our opinion, would maximize the benefits students could gain from these types of research projects, and conclude our study by proposing some future research ideas.