Sunday, November 29, 2015

Introduction to Differential Calculus for Algebra Students


My earlier blog about math education generated some offline discussion. In it, I suggested that students who are stuck on fundamental concepts should not continue forward until they understand those concepts. I also suggested that students could study principles of calculus while studying algebra. These two points seemed to some readers to generate a contradiction. I disagree. All that is required to study an advanced math topic is the set of specific fundamentals foundational to that topic. Beyond that, the traditional schedule followed in K-12 curriculum is arbitrary. An algebra student who understands how to plot a polynomial and how to find the slope of a line is prepared for differential calculus in its most fundamental form. Thus, for example, geometry and trigonometry are not fundamental to basic calculus. On the other hand, a student who does not understand the concept of slope should master that concept before attempting basic calculus.

As an illustration of how an algebra student could learn basic calculus, I include a slide show (below). Students working through the example should not simply read the slides, but should reproduce the ideas on paper and work out the algebra for themselves. Upon completion, they should understand how to find the slope of a line tangent to a continuous function, which represents the derivative of that function at that point.

Note the missing signs on slide 19: Students should take advantage of this error by filling them in themselves, then checking their result against the next slide.