Calculus is a specified field of mathematics. Limits, derivatives, integrals, and infinite series are all applied to various types of functions (i.e., linear, quadratic, exponential, trigonometric, etc.). They each depend on the understanding of basic arithmetic as well as specific concepts of other branches of secondary level mathematics such as algebra, geometry, and trigonometry. Prior to taking a calculus course students should develop specific skills in each of these areas of study.
Below, I briefly discuss key skills and concepts required from three branches of mathematics (algebra, geometry, and trigonometry) to prepare students for calculus. I discuss this material as it relates to differentiation, a fundamental component of calculus.
Differentiation converts formulas into other formulas. These formulas are functions. Algebra is the study of functions. Since functions are the object of differentiation, it is imperative that students entering into a course of study in calculus have a thorough understanding of functions.
Many secondary students can follow instructions to substitute a value into an equation, but the study of functions will provide a greater understanding of why the method of substitution works and why a relationship exists between the parameters.
The result of differentiation is a derivative. Derivatives are used to calculate the slope of a line tangent to a point on the graph of a function. A student can find an estimated value using the slope of a line secant to the same graph, but calculus makes it possible to find the exact value. Students must have a thorough understanding of graphs, slopes, and functions to understand the use of derivatives with respect to graphs of functions.
Trigonometry (Angles, Periods of revolution)
Trigonometry is the study of trigonometric functions. Trigonometric functions are used to calculate measures involving triangles, periods of revolution, etc. During the course of a calculus class students will encounter problems involving trigonometric functions which are useful for explaining periodicity. Students studying calculus should understand trigonometric functions, their properties, and their graphs.
As discussed above, students preparing to take calculus should have a specific set of skills relating to calculus content. The major subjects required for the development of these skills, and the understanding of the related concepts, are algebra, geometry, and trigonometry. More specifically, students need to have a thorough understanding of functions, graphs, slopes, and trigonometric functions, to name a few, in order to effectively prepare to enter into a calculus course.
Although students have the benefit of the use of technology to solve many problems, they must have an understanding of the relationship between the functions used, their graphs, and the derivative so they can understand the output generated from the use of technology. Simply plugging values into a technological tool does not provide this level of understanding. In most cases, students will not have access to technological tools in college level math courses (see “Calculators Not Allowed” on this blog site), so it’s best to learn and understand the concepts without these aids.
Do you have any other ideas about concepts students should learn prior to taking a calculus course?
- Graphing Functions: By Hand versus Using CAS… (mathequality.wordpress.com)
- Algebra Bootcamp in Calculus (samjshah.com)