A visual demonstration of the derivative as the linear approximation of the rate of change is an excellent suggestion. There are several ways to create such a graph in Maple. The Explore command applied to a plot command is one of them. The interactiveparams command in the plots package is another, but both these alternatives require knowledge and use of syntax. A syntax-free (or clickable) way to build the same animation is through the Plot Builder, launched from the Context Menu. Selection of the animation option or the "Interactive Plot with 1 Parameter" option would lead to equivalent visuals.

RJL Maplesoft

@Alex Smith 

Yes, its just a different philosophy of learning. My experience in the classroom (35 years) convinced me that I best served my students if I helped them to avoid mistakes. I always found that I had the hardest time learning something I had previously gotten a wrong impression of. Unlearning is hard, so I always tried to get my students to "get it right" right from the beginning.

Just a different perspective on what works best for both student and instructor. But it's good to be reminded that others have different views, and that these differences for the most part need to be respected.

@Alex Smith 

Let's see if I correctly understand this comment. Does it suggest that the chance for cognitive conflict is diminished with a tool that makes it easier to make the exact same mistake described in the example?

Suppose the student wrote the same wrong expression for the tangent line, and then added it to a graph of the function via drag-and-drop instead of via some more complex syntax. The same cognitive conflict would arise, only it would happen faster and more easily with "syntax-free" computing. And with syntax-free computing, the student wouldn't have to check out the detour "Is my syntax correct, or is the error in my math?"

Bottom line: I don't see that ease-of-use in any way reduces the pedagogical opportunities afforded by cognitive conflict. Am I missing something?

@Markiyan Hirnyk 

The subject areas touched by the 44 problems posted so far are those that the "typical" college student in math and science programs meet in their first two years of study. The point of the work is to show both a pedagogical approach and to demonstrate how to implement this pedagogy in a syntax-free manner.

There are more than 100 additional problems in the pipeline, and as of this moment, more than half that have now been recorded and are being readied for posting. But initially, the six subject areas themselves will be augmented with problems only from precalculus math (algebra and trig).

Beyond that, where the "clickable" paradigm works, problems from additional areas can be considered. But it would be useful to understand that the primary goal of this collection of activities is to demonstrate an approach to the use of technology in teaching and learning college-level mathematics, and to show that the ease-of-use techniques in Maple that flatten itslearning curve make it possible for this pedagogy to be implemented in a syntax-free manner.

Maple is certainly a useful tool in many other subject areas of mathematics. The pedagogy advocated in the problems so far posted is certainly applicable, and has already been demonstrated in my Advanced Engineering Mathematics ebook. However, not all of what's in that volume is easily rendered in a syntax-free form. Perhaps there is need for a separate web area reserved for problems in which the "clickable" paradigm is less well-developed.

By the way, a number of improvements in the syntax-free environment of Maple have come from pushing the boundaries of what could and could not be captured in this format. So, perhaps one outcome of extending this project to other subject areas will be the augmentation of the syntax-free tools in Maple. Is this something our user-community can contribute to?

@Axel Vogt 

Axel,

I agree that a worksheet full of 2D math and slated for Application Center, makes a poor post to MaplePrimes. Henceforth, I intend to do what you suggest - summarize for Primes and leave the worksheet for download.

As to "Why can't I have..." I must tell you that I was hired by Maplesoft to reflect the perspective of the classroom user of Maple, not as a programmer. I have little influence on the actual code in Maple. I make recommendations, I ask for things, and even sometimes resort to begging and bribing. I, as much as you, would like Maple to be as intuitive as possible. I'm not the one to answer your second question. But don't stop asking it. Maplesoft does say it listens to its users.

RJL

@Torre 

Thanks for the pointer to the DG worksheet. I'll have to study that for a bit. My 1970 PhD thesis was in relativistic cosmology, so at one time I knew some differential geometry and continuous group theory, but that was a long time ago.

I've looked at the DG package, but there's so much in it that I'm not sure I'll ever master it. Every bit of digging helps, though.

RJL

@Torre 

It would certainly be useful if this comment were expanded to a full demonstration showing how one might use such a basis in the DifferentialGeometry package. For sure, I would appreciate being shown how.

RJL

Simply work in the Student LinearAlgebra package where all quantities are assumed to be real. Also, the VectorCalculus packages do not conjugate for dot products.

RJL Maplesoft

I've looked at the two worksheets supplied in this question, but they import data from Excel files that weren't provided. I've privately requested access to the data. If I get the data, I'll explore the calculations to try finding out why there's a problem. I'd like the conversation to be off-line until I have something to report. Meanwhile, if anyone else can spot what's happening in the failed calculation, please enlighten us all.

 RJL Maplesoft

Unfortunately, Gem 12 contains a typo that leads my discussion into error. The typo appears in the definition of LEFT, the left side of the equation being investigated. This left side is the product of two radicals, and under the second radical I typed the number 1 where I should have typed the imaginary unit "i". I used the resulting graphs to deduce an erroneous conclusion. The correct conclusion is obvious from the (colorful) graph provided by Alec Mihailovs.

I've revised the worksheet upon which this blog is based, and will replace the download with the revised copy. Sorry for that, folks. And thanks to Alec for sparking my second look at that Gem.

RJL Maplesoft

I found Robert Israel's investigation of guessgf interesting and puzzling. It was interesting to see that ultimately, the limit of the sequence of Picard iterates was found by solving the very differential equation we started with. It was puzzling because it raised the question: What math is Maple implementing to recover the ODE from whatever information is in "L"?

Unfortunately, I could not see where either he or I had defined a quantity "L" in that Gem. Robert, what information was in "L" and have you any idea what mathematics it takes to go from L back to the ODE?

RJL Maplesoft

frem(75,evalf(2*Pi)) => -0.39822370, which is 75-12*evalf(2*Pi). On the other hand, 75-11*evalf(2*Pi) = 5.88496161, so is the problem of subtracting multiples of 2 Pi solved yet? I guess it depends on what the user actually wants.

RJL Maplesoft

frem(75,evalf(2*Pi)) => -0.39822370, which is 75-12*evalf(2*Pi). On the other hand, 75-11*evalf(2*Pi) = 5.88496161, so is the problem of subtracting multiples of 2 Pi solved yet? I guess it depends on what the user actually wants.

RJL Maplesoft

 Thanks, Alejandro. I appreciate your continued development of this topic.

RJL Maplesoft

 I am edified by the comment demonstrating the use of rules and the applyrule command. However, the rules for sine and cosine are quadrant-dependent. This approach is useful when the quadrant of the angle is known. Otherwise, a sign error could be introduced.

But the illustration of the use of applyrule is quite valuable and has been added to the Little Red Book.

RJL Maplesoft