Have you tried teaching calculus where everyone gets to use Wolfram Alpha and/or Mathematica? I did for three years.
I discovered that the students really don't understand the concepts and more importantly don't want to. They just want to mimic problem types. They don't want to understand the why. Plug and chug is their true desire.
Using the computer exposes right away if a student understands the concepts. You can't get started on a problem if you don't know what to tell the computer to do or why. I went back to the old paradigm. A few students got it but most never understood.
> I discovered that the students really don't understand the concepts and more importantly don't want to.
The problem is that you are forcing people who don't care about it to learn it. Ofc they'll take shortcuts.
> Plug and chug is their true desire.
I mean, most tests are very plug and chug so it's no surprise.
Also note that it might be unreasonable to expect students who never used these tools to use them for the right purpose. Mathematica is a complex tool but I can't imagine that much class time is dedicated to the tool itself.
Actually it's not a complicated tool with the free form input, integration with Wolfam Alpha, and templates for the commands used. Also during every test I would correct any command that didn't execute properly.
Your point originally was that it requires too much work on the part of teachers/administration to update the curriculum for 21st century. I'm pointing out that this view while true in some cases may not be true in the present situation. It's easy to ascribe to laziness why things are done in education the way they are to someone not involved in teaching.
Sounds like Mathematica is the way to go then? Because it turned out students who could do the problems. The 'old paradigm' leaves most behind, which serves no one.
Students could not do the computer based problems. Save for a few. More students get through with the old paradigm.
Old paradigm problem. Here is f, using the definition of derivative find f'.
New paradigm problem. Here is f. Here is a Mathematica function defining f(x+h) - f(x) divided by h. Graph this function for appropriate values of x and h to show whether or not f is differentiable at 2.
The old paradigm problem you just proceed as in all the examples. In the new problem they get intimidated because it involves using a computer and not a graphing calculator. They don't understand that you keep x fixed at 2 and vary h around 0. When they start the problem they give nonsensical input to the computer, get nonsensical output and promptly blame the stupid program.
To find out whether f is differentiable at some point, I'd naively draw f around 2 and look for obvious discontinuities. Why encrypt the problem in such a way? Alternatively, finding f' and explaining if it is undefined at the point in question would work even more reliable - for corner cases the visualization of a graph might hide the fact, i.e. if the resolution is too poor.
Regarding the first sentence your wrote. That's how you solve the problem. Graph the difference quotient for x=2 and h in [-0.1, 0.1] or some such suitably small region. It's quite easy to do in the computer. One just needs to know that this is what you need to do. I did not encrypt the problem. It's quite straightforward provided one knows the concepts.
If students are allowed to use Mathematica during a test then asking them to find f' using the definition of f is not helpful in determining if they understand the concepts. They, for all reasonable problems, just need to execute a single command:
Limit[ (f[x+h]-f[x])/h, h->0]
This doesn't really test their understanding. If one is going to allow students to use Mathematica on a test then the problems need to be adjusted.
I see. Then I don't understand what point you were trying to make. There are lots of different ways of testing understanding of a concept. I presented one way of doing so with students having access to a computer.
The point of the problem I gave was not to find the derivative or even to know if a function is differentiable. The point is to test their understanding of the definition of derivative. It's a good problem in that is tests whether a student understands that in the two variable expression we call the difference quotient one of the variables is fixed for purposes of the definition. The students need to know that it is h that varies and not x. This is an unusual occurrence and requires some getting used to.
I discovered that the students really don't understand the concepts and more importantly don't want to. They just want to mimic problem types. They don't want to understand the why. Plug and chug is their true desire.
Using the computer exposes right away if a student understands the concepts. You can't get started on a problem if you don't know what to tell the computer to do or why. I went back to the old paradigm. A few students got it but most never understood.