Essay - Understanding the Concepts of Limit and Continuity in Calculus Courses...

Understanding the Concepts of Limit and Continuity in Calculus Courses at the High School *****/or Early College Level

Background and Overview.

***** need for a solid background ***** mathematics for high school and college students today is well documented. According to Conaway and Rennolds (2003), "With the onset ***** the technological age, students must complete as much math as possible during high school. Mathematical skills are essential to gaining access ***** college and pursuing a c*****er in a *****-, science-, or technologically related field. A student's mathematical ability can be a determining factor in choosing a c*****reer" (p. 218). In fact, the 10 fastest growing career *****s include five that are computer-related ***** rely heavily on ma*****matics knowledge; these ***** fields include computer engineers, computer support specialists, systems analysts, database administrators, and desktop publishing specialists so clearly, young people who c*****tinue their pursuit of higher mathematics throughout ***** high school and ***** years will enjoy additional career opportunities (***** & Reynolds, 2003).

Any experienced teacher, though, can readily testify that actively engaging young people in the learning process can be a challenging endeavor under the *****st circumstances, and the rigors involved in acquiring a solid foundation in algebra, calculus ***** other higher mathematics can be daunting ***** many young learners to the extent ***** *****y either perform poorly or withdraw from the coursework entirely. In fact, according to Cipra (1988), "The teaching of calculus--when ***** do it, and even why to ***** it--has become a major issue in ma*****matics.... Calculus reform is considered urgent because of the course's command*****g position in the early ********** experience of students hoping to go on in science, engineering, business, and other fields.... Yet at some institutions as ***** as 50 percent ***** the students enrolling calculus either fail or withdraw from the course" (emphasis added) (p. 1).

Clearly, *****n, helping high ***** ***** early college students achieve mastery of complex subject m*****ter such as calculus frequently requires more than a cursory understanding of how young learners acquire and retain knowledge ***** what *****es serve to facilitate this acquisition and retention. For instance, in their study, "Conceptual ***** of introductory calculus," White and Mitchelmore (1996) point out that, "Research into the ***** ***** ***** h***** s*****n a *****le spectrum of concepts that cause problems for students. In particular, student difficulties with the abstract ***** of rate of change ***** function are ***** documented. These concepts involve mathematical objects or processes specific to calculus. Another aspect that needs to be considered is the question of ***** other ***** are involved in applying ***** knowledge" (p. 79).

While there are some concepts that are specific to calculus involved, there are ***** aspects of delivering these educational services that can draw on the existing curricula re*****m efforts, and many of these have been applied to the ***** classroom already. According to Ferrini-Mundy ***** Graham (1991), "Much ***** the calculus reform discussi***** has centered on streamlining ***** traditi*****al syllabus to create a 'lean and lively' course" (p. 628). Besides