Archive for the ‘CalculusMV’ Category

Kalkulus Shares Animations

Today’s guest blogger is Kelly Liakos, a Calculus instructor for 20 years, who shares his site of Calculus Animations with us.

In 1996, the Math Department at Santa Fe Community College (where I taught) decided to add a technology component to its curriculum, which is the point at which I began developing computer animations and computer labs for my courses.

There are basically four ways in which components on my website have been used.

1. In my classroom I had a computer and a computer projection system. I incorporated the animations in my lectures so that while discussing theory I would use the computer animations as examples as we go. When I first started developing animations my theory was that Calculus is dynamic–before the advent of the technology it was difficult to get this point across. However by actually seeing Calculus in motion as you explain the theory makes it much easier to grasp the concepts.
To get some idea of how this works see the page Calculus 1 – Limits and Derivatives. On this page the discussion is very similar to the discussion I used in lectures, with the animations being shown roughly in the order presented. Allowing for other examples and questions from my students that one page encompasses approximately four lectures.
Another good example is the page on Flux Integrals and the page about Graphs of Trig Functions.

2. At Santa Fe Community College there is also a Computer Lab component. On my Computer Lab page you’ll find several examples of computer lab projects (for Mathcad) that the students do. During lecture, the student has seen the theory, watched the animations, and now in the computer lab the student gets hands-on experience working in groups of 2 or 3 to apply what they have learned. In some cases this serves to clarify what is they are supposed to have learned.

Initially, I spend a lot of time answering questions to make sure the student is familiar with Mathcad, but as time goes on the students do it without my supervision. In fact, many of the lab assignments are assigned as out of class projects with no supervision at all.

3. The animations and notes also serve as an out-of-class resource for the students to view (as little or as much as needed) to reinforce what they might have missed in lecture. As the resources are available it is the next best thing to actually recording lectures.

4. To a lesser extent you’ll notice that in many of the notes, mechanics is discussed very little.

While we do spend time in lecture discussing techniques of differentiation and integration, with the reform movement, the focus is much more on understanding the theory and applications. In many of the applications once we have the set up and explanation we let the computer do the mechanics.

Of course the extent to which an individual instructor stresses the mechanics varies widely and I’ll not even begin to discuss this here. Basically my theory is that when you combine the theory and technology you have Calculus at its most powerful.

The reason I decided to stop teaching at this time was so that I could concentrate on developing animations and the corresponding notes. My website contains my ideas on my various courses but I would also be interested in developing animations to supplement other professor’s ideas as well.

Thanks Kelly for the post! To find Maria in India, go here.

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Math Provides Beauty and Truth in Physics

Murray Gell-Mann gives a TED Talk entitled Beauty and Truth in Physics.

The second part of his presentation is subtitled “Math Matters” (you can forward to this one)

Quote from Math Matters: “We express these things mathematically, and when the mathematics is very simple… when, in terms of some mathematical notation you can write the theory in a very brief space, without a lot of complication, that’s essentially what we mean by beauty or elegance.

The third part of his presentation is subtitled “Symmetry Matters” (again, you can forward to this)

As our notation improves and we are able to incorporate symmetry into equations, the equations become simpler and more elegant. Here is an example (from the talk) showing the progression in the equations for Relativity – I think that a multivariable calculus class would probably be able to appreciate it best:

This was a great little tidbit – a quote from Newton on why he was not mentioning his theory of gravity in one of his books:

Newton was worried that he would be labeled an “extravagant freak” and that readers would thus dismiss the rest of the book.

Best quip of the talk – Newton could have really written a great essay on “What I did on my Summer Vacation” (referring to the time that Newton spent away from school during the plague years – some of the most productive time of his life).

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Mathematical Visualizations for Multivariable Calculus

Jonathan Rogness (UMN) has a great collection of mathematical visualizations for multivariable calculus that run on Java. I don’t teach multivariable calculus (and hope never to have this experience), but I always love looking at the 3-D surfaces.
You may remember Rogness from the Mobius Transformations video I blogged about a while ago.

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Flash and Math

Barbara Kaskosz and Doug Ensley have put together a new site for learning to use Flash & Math with all their tutorials (beginning, intermediate, and advanced.

To see some of the Flash applications that have been built, go to the Math DL site and search for Flash/Shockwave resources (as shown in the image below).

Amongst other things, there is a nice collection of tools for multivariable calculus developed by Barbara.

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