Exploring Functions with Calculus and Dynamic Software

When I took calculus as an undergraduate, it was presented in a huge lecture hall with hundreds of students and I was well aware that it was known as a “weeder” class for engineers, Pre-Med, and other science majors. I distinctly remember always feeling like calculus was a subject with numerous well-guarded secrets that most were unwilling to openly state. 

Ways to Reduce Math Anxiety

Even though I earned A's throughout high school in math and science, my college calculus class caused me to experience my first twinges of math anxiety.

Recently, I took an online class called Teaching and Learning Calculus With Geometer’s Sketchpad. Not only did I find my anxiety towards calculus lessen, but I also better understood the basic concepts and ways to translate them to students. I found a way to teach the subject that doesn't involve an impersonal lecture hall and hundreds of people, but relies on a personal approach using dynamic software, presentations, and  projects.

Derivatives

One important thing I remember from my undergraduate calculus class is how to find the procedural derivative by using the exponent of each term coefficient and then reducing the exponent by one, and how to do the opposite of this to find the integral.

So, for a concrete example, let’s say you have a simple polynomial such as 4x^2+6x. First you would take the 2 down from the first term and multiply it by the 4 as the coefficient. Therefore, the new coefficient of the first term would be 8. You then reduce that exponent by 1. So the derivative of the first term is 8x and the derivative of the second term is 6. The entire new derivative equation would be 8x+6.

With Geometer’s Sketchpad, students can actually move a secant line upon an original function to see how the slope of that secant changes as it moves along that original function. With such visual and dynamic software, they can also see how they can graph all the slopes of that secant to produce a derivative function. Furthermore, we can physically get a feel of how a more accurate rendering of the derivative is possible as the secant line decreases infinitely to the tangent line. Perhaps most important pedagogically, highly engaging visual projects can be created with such dynamic software, with an emphasis on presentation and revision rather than just giving a one-time midterm and final.

Integrals

For the integral, we apply integration which is the opposite process we used for finding the derivative . . . along with adding the ominous “C” to the equation. For instance, if we have the function 3x^2, we then add on to the exponent and remove the coefficient so that there becomes infinitely many other anti-derivatives such as: f(x)=x^3+C, where C is the constant of integration.

In using dynamic software in connection with integrals, students can more easily relate the idea of an area function to that of a definite integral of a function. For instance, moving along a definite portion of the function f(x)=2x, students can see how and why taking the area under that function produces an upside down parabola. The software also allows for a better understanding of a negative area and equal areas that cancel out to become a net of zero.

Anti-derivatives

I didn’t learn the concept of an “anti-derivative” twenty years ago, but with the help of dynamic software and Fetter, I learned how multiple undefined integrals may be produced from a function. The software shows the user how the same anti-derivative shape can be translated upon the y axis, thus the graphical importance of the “+ C” component to the anti-derivative equation. 

Projects with Revisions

Another aspect of teaching calculus, which may be applied to teaching generally, is the need to put an emphasis on the process, rather than the final product. Students should be invited to create videos of what they have discovered about calculus using dynamic software and then have the instructor give feedback on what they did well and what they could work on in a revision. Students should also be encouraged to write about their learning process in constant reflection exercises. They might also be challenged by being asked to describe how they would teach the material, and identify where confusion still remains.

Calculus Lessons:

Seeing is Believing: Derivatives

Students take the derivatives of functions to solve them. Students examine the relationship of the graph of a derivative to that of an original function. This lesson provides a hands-on way for students to work with derivatives.

Local Linearity

Students graph various functions on a TI-calculator and zoom in to identify how the slope of the function changes. This is an interactive means for students to better understand functions.

Maxima and Minima Problems

Students determine the maxima and minima of quadratic equations. Students also use what they have learned about derivatives to determine the maxima and minima in real-life applications.

The Area Function

Students explore the concept of area under a curve. In this area under a curve lesson, students find integrals of various functions. Students use a Ti-Nspire to graph functions and find the area under the curve.