The topic I chose is calculating the area under a curve in calculus. The prerequisites for this topic are algebra and some calculus.
Prerequisites are the topics you need to understand before tackling a new topic because it uses those as a foundation from which to build. In order to learn how to calculate area under a curve, students need to have a deep understanding of functions, know how to manipulate functions algebraically, and understand the concepts of limits. It can be really difficult for students to learn calculus without a strong background in algebra, since calculus frequently requires skills such as combining like terms, simplifying, factoring, etc. If students get tripped up at those steps, it takes away from focusing on the calculus, rather than the algebra. Additionally, a deep understanding of limits is essential to understanding calculus.
Technology can remove these prerequisites by giving visual demonstrations of how to fund the area of a curve. Students don’t need to know all the specifics of algebra to be able to understand the visual of a series of rectangles under a curve as an approximation of area. Technology can also remove the prerequisite of limits since a graph could be animated in such a way that it demonstrates the concept by starting with a few rectangles, then adding more and more. This shows how the more rectangles you use, the closer your approximation is, and with this visual, students could probably make the mental leap to the idea of shrinking the rectangles’ widths down to nothing.
In that same vein, technology could also be used to demonstrate the concept of limits using Archimedes’ method of exhaustion to approximate pi. Students could use a circle with inscribed and circumscribed polygons to understand that one overestimates the circumference and one underestimates. Technology would allow students to watch as the number of sides increased to get a very good visual demonstration about how the true value is between the perimeters, and the more sides the polygons have, the closer it is to the true value.
I like the idea of younger students getting some exposure to calculus before completing the required sequence of courses because there are a lot of results and applications that would make material more interesting and relevant for them. I think that a whole curriculum without prerequisites might work for students who don't intend to pursue math education. A lot of the standard math curriculum is designed to prep students for calculus and a lot of students will never take calculus. I think there could be some interesting and meaningful math classes where students learn about a variety of topics without necessarily getting the traditional prerequisites for each. However, I wouldn’t want to remove all prerequisites from calculus since those skills are essential for students who intend to pursue advanced mathematics education.