This is my response to this post by Mr. Morford concerning his view of Wolfram|Alpha.
In my Finite Math class this semester, I have been using Wolfram|Alpha a lot. The other day one of my students wanted to use Wolfram|Alpha to reproduce one of the feasible regions, when all she knew were the corner points. I told my student to do it directly. Our first query was 'shape with corner points at (1,3) (4,5) (8,6) and (3,1)'. Wolfram|Alpha didn't take very well to that.
After a few trials, I put in the query: 'polygon with vertices at (1,3) (4,5) (8,6) and (3,1)' and poof, Wolfram|Alpha gave me the required feasible region. My student immediately commented that there's almost no way without my guidance that she would have ever thought to use the words polygon and vertices. But she did tell me that she now recognizes the importance of knowing what things are called and the vocabulary that goes along with mathematics.
I think that this part of the roadblock to some students learning mathematics — that they simply don't know what things are called because they have somehow been taught that mathematics is simply about getting the right answer, and if you can't get the right answer on the first try, then you simply just don't know how to do math at all and you never will know how to do math. Because math is difficult.
But take this first example of Wolfram|Alpha, our first query wasn't correct, so we tried until we got a query that did work. We didn't give up on one try. My student didn't say to me, 'Oh well, we couldn't get Wolfram|Alpha to do this on the first try, so Wolfram|Alpha must be hard to use, and we will never know how to use Wolfram|Alpha." No, instead I saw the same phenomena that I see with many of my students — once they see that Wolfram|Alpha, or any other computer-based tool can do a math problem, they are willing to figure out how to get that tool to do that math problem so that they don't have to do it.
However, if we look at this from a different perspective, this may actually be a good thing. First of all, in the long run, the students are still doing the math. The students are getting an answer. And if Wolfram|Alpha is doing the computation, then we have more time to focus on the critical thinking. And second of all, isn't the whole trial and error step of how to get Wolfram|Alpha to read my query part of the critical thinking process that the students would have missed had they not used Wolfram|Alpha? If I see a student trying different queries until he or she figures out what is an appropriate query, that makes me happy. That idea of investigating as a part of the learning process is something that I think that is missed with traditional, non-computer learning.
Think about it this way. In a traditional classroom, if a student who is solving an equation tells me that they cannot figure out the next step, I generally will ask the student what they think the next step will be. Now, with Wolfram|Alpha, I can say, 'Enter the equation as a query in Wolfram|Alpha," and now you tell me what the next step is. Then I ask the student to enter another, and another, and yet another. And eventually through investigating how different equations are solved, the student starts to get a better sense of what it is that you are allowed to do when solving an equation, and what it is that you are not allowed to do when solving an equation.
Just a thought, whether the problem is difficult, or easier, I see plenty of open opportunities for students to use Wolfram|Alpha as a tool for investigative learning. But I definitely do believe that the students still need the instructor's guidance to lead their investigations and their critical thinking.