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6. Conclusions

 

As a result of our research, we believe we have learned two important lessons. First, in studying the history of the Calculus, we became acutely aware of our own lack of understanding for the subject. Second, the software that automates algebraic manipulations requires in itself a complex set of theory, and we have gained an appreciation for this tribute to the accomplishment of modern technology. Despite having been trained and performed well in Calculus exams and courses, it is clear that our appreciation for the subject matter was minimal at best. We believe that studying the history of the subject has enhanced our understanding, yet there clearly remain numerous gaps in our knowledge.

As students, we have been trained not to understand the origins and reasoning behind the science we study, but merely to be adept at applying memorized techniques to problems. Though the application of science is undoubtedly a useful skill, we wonder whether we ought to be brought through the development of the subjects we study. Calculus is merely one branch of science, is it possible to truly appreciate subjects such as Physics without studying the experimental processes that ultimately converted popular thought to heliocentricity and universal gravitation? What, after all, do we really understand about the knowledge we purport to have attained if we do not understand the process that led to its development? Though studying the history of a subject is not a necessity in understanding its development, it seems that merely learning science as an act of route memorization is inherently contradictory to the process of science. After all, the process of science requires that one not accept theory without evidence, that one should attempt falsification within reason, and demands that one not take a theory as truth because it is argued from authority.

This paper was an attempt to remedy our own lack of understanding of the origins of one science we had studied. Though there still remain topics in the history of Calculus that we were unable to research or fully comprehend, some of the holes in our understanding have been filled in. In addition, the development of our own Computer Algebra system taught us the strength in learning through experimentation. Facing the technical issues head-on brought us a far deeper appreciation than any amount of reading could have conveyed. The major lesson learned is that our best approach to understanding is not to simply research known methods, but to simultaneously use and understand those techniques.


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