What’s the best way to teach Calculus?
Dr. Del explores the modern approach to teaching Calculus in his Special Report, Calculus-A Modern Approach. You can read the Special Report below or click on the title link to download a PDF.
A Modern 21st Century Approach (since 2009)
Craig Hane, Ph.D. aka Dr. Del, Founder, Triad Math, Inc.
What are the math problems Calculus aims to solve?
Let y = f(x) be a real valued function defined over a domain (a,b) where a and b are real numbers, or – or + infinity.
I. We want to be able to graph this function by finding its:
x- intercepts (roots where f(x) = 0) and y- intercepts (f(0))
Maximum points and Minimum points
Points of Inflection
Asymptotes – Vertical – and where x approaches + or – infinity
Plot the function in detail
With this information we also will be able to see where the function is increasing and decreasing, and its concavity. We might want to be able to determine these two things at a particular point without graphing the function, or its instantaneous rate of change at a point.
Algebra and Differential Calculus are the tools utilized to accomplish all of these objectives, classically, (usually called Calculus 1).
Today we can do all of this quickly and easily with a modern tool like Wolfram Alpha. Indeed, we can do this for functions which were essentially impossible to deal with using the classical calculus tools.
Of course, Wolfram Alpha will also give you the derivative and anti-derivative of any function, including Special Functions if you desire them for some reason. It will also give you higher order derivatives if you so desire.
Wolfram Alpha will give you the roots of any polynomial of any order including the complex roots. This is virtually impossible to do with higher order polynomials with classical algebraic techniques.
Then there are many topics covered in Calculus 2, and 3.
II. We want to find the area enclosed by the graph of the function from x = a to x = b, and the graph of the absolute value of the function from a to b. This utilizes the “definite integral”.
This was classically accomplished using the fundamental theorem of calculus and anti-derivatives using what are called “the techniques of integration”. This can be quite difficult and often impossible. And, when that fails we must use numerical estimations.
Today, Wolfram Alpha does all of this automatically very quickly and easily, and will do it for functions that are essentially impossible to deal with classically.
III. We want to find the “arc length” of the graph of the function from (a, f(a)) to (b, f(b)).
Classically, this involved setting up the proper integrand and then finding the definite integral. Many times it was impossible to find this integral using classical integration techniques and ordinary functions. The arc length of an ellipse is a good example of this. It involves what is called an “elliptic integral”, which has no anti-derivative using ordinary common functions. So, the Fundamental Theorem doesn’t work here. This is often the case. And anyway, quite often anti-derivatives can be very difficult to find.
Today, Wolfram Alpha solves this problem very quickly and easily.
This also applies to curves defined parametrically.
IV. We want to find the volume or surface area of a solid of revolution when the graph of f(x) is revolved about either the x or y axis.
Today this is done automatically and easily with Wolfram Alpha.
Classically, one had to set up the appropriate integrand and then perform a definite integral calculation. Both of these tasks could be difficult and, sometimes impossible.
V. We want to find a polynomial, P(x), approximation of f(x).
This involves utilizing a Taylor infinite series expansion which is truncated to yield the P(x). This requires calculating the higher order derivatives of f(x) to the desired order of P(x). Sometimes this is easy, but often it is difficult. It is always time consuming, boring, and error prone.
Wolfram Alpha does this automatically, quickly and easily.
VI. We want to find the derivative of a function which is implicitly defined. This is used, for example, to find the derivative of an inverse function, or to handle related rates situations.
Classically, this is not too difficult since it involves only simple rules of differentiation such as the Chain Rule and the Leibniz Rule, but it can be algebraically messy.
Wolfram Alpha does it automatically. No muss. No fuss.
VII. Find the equation of a tangent line to a point on a parametrically defined curve. Sometimes this can be done classically, but often it is difficult to even find the points on the parametrically defined curve.
Wolfram, does both of these tasks quickly and easily.
VIII. If you can think of anything else you want to do with calculus, I am confident Wolfram Alpha can do it. It is an extremely powerful tool. Calculus is just scratching its surface.
One can learn to solve any STEM calculus problem by using Wolfram Alpha in a few weeks or leisurely months by mastering the appropriate commands for Wolfram Alpha, compared to the typical challenging one to two years for the classical approach.
The classical tools taught in calculus are marvelous and were historically extremely important. After all, they are responsible for our modern civilization! But, they were difficult to learn and master, and quite limited in the problems they could actually solve.
The same can be said about logarithms and the slide rule for arithmetic calculations, or manual carpenter tools, or the horse and buggy, or old fashioned dentistry or medical tools.
Today, these tools are obsolete.
The classical tools often are much more difficult to learn and master than the modern tools. In the case of calculus, “techniques of integration” may have caused more students to abandon a STEM course of study than anything else.
These “techniques” are now obsolete, having been replaced by much superior 21st century tools like Wolfram Alpha.
Whew. About time!
What a modern STEM professional needs today is to master the use of a tool like Wolfram Alpha. In addition to calculus, the STEM professional needs to understand many more things like differential equations, linear algebra, matrix algebra, eigenvalues and eigenvectors, two dimensional surfaces from multivariable calculus. Things like DIV, GRAD, and CURL. Again, Wolfram Alpha to the rescue. This is the modern 21st Century way. Unbelievably better!
Of course, Wolfram Alpha is based on Mathematica (now free in the Raspberry Pi) and Mathematica does much more. It is a powerful programming language with immense capabilities.
A student probably is best served by learning Wolfram Alpha first, and then learning the new Wolfram Language (released in 2014 on the Raspberry Pi) and Mathematica.
I might add that Wolfram Alpha and Mathematica also help a student learn theoretical mathematics too, thanks to its powerful experimental interactive power, plus its access to vast databases of mathematical knowledge and functions and operations. These tools are so good, they seem like magic, to me at least.
NOW, your student can master the mathematics necessary for any STEM career in about one year after “precalc”, which itself should take about a year or two post-elementary starting with the Practical Math Foundation.
AND FINALLY, OUR INVITATION TO YOU…
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