Why do Socratic users not just use WolframAlpha?
Another reason why folks use Socratic: together, we're creating a lasting artifact.
By answering questions step-by-step, we're creating a (free) learning resource that helps the student who asked, yes, but that will also help any student who has that same question in the future.
There is an argument to made for just giving the student a solution so that they can work backwards from it (Wolfram style)—and we should all encourage students who ask questions on Socratic frequently to try answering questions themselves for this very reason.
However, by making thoughtful, step-by-step solutions available on Socratic, we're helping thousands of students in that first moment of confusion; that powerful moment of learning that makes them feel like they can conquer this problem, this concept, this subject.
Because it gives you an answer (might not be the one you're looking for) that works, but does not tell you how to get there unless you get a subscription AND bother to look yourself.
And you know what, the point of Socratic is to teach the process of getting to the answer, not giving you the answer without an effort to teach.
On the other hand, if you have to do a seriously complicated problem and you simply can't figure out how to do it by hand (or shouldn't know how to do it by hand, even), definitely give Wolframalpha a shot.
HOW WE MIGHT INTERPRET WOLFRAM ALPHA
There are some other reasons for not using Wolframalpha right away, even for something relatively simple, including validity and scope.
In general, if you use Wolframalpha, since it is programmed by human beings to be automatic, it doesn't think like a human being. It doesn't incorporate the nuances of a human being's brain, but instead tries to do problems by following a set algorithm.
- If it says you're wrong, then you have reason to doubt that you're correct. It's pretty good at being right about being wrong!
- If it gives you an answer it says is correct, it may be wrong by the standards of your classroom, or it may require math that you don't know.
For instance, if I were to use Wolframalpha to check this mathematical query:
#color(green)(int 1/xdx = ln|x|)#?
...then I would see that it confirms I am correct for
That is acceptable in the classroom. But if I check this mathematical query instead:
#color(red)(int 1/xdx = lnx)#?
...it returns that I'm correct no matter what domain I use, when that wouldn't (or shouldn't) be accepted in the classroom.
I believe that maybe the reasoning for the proper domain is left to the asker, but I would guess that the asker would attempt to paste the answer into an online answer box without questioning the validity of that answer. That's not a good habit...
IT GIVES WEIRD ANSWERS?
Another example is getting something crazy or weird for an answer that may not be expected for an answer.
#int 1/(sqrt(1+x^2))dx = sinh^(-1)(x) + C#
...okay, I mean, it's right, but who actually remembers that? I don't.
A more common or intuitive way to achieve the answer in a classroom is to use trig substitution.
#=> int 1/sectheta * sec^2thetad theta#
#= int secthetad theta#
#= ln|sectheta + tantheta| + C#
#= color(blue)(ln|sqrt(1+x^2) + x| + C)#
And if you ask whether the derivative of this answer is equal to
I find wolframalpha (without subscription) useful like an advanced calculator. It's good for checking calculations, but there are some things to watch out for.
By default cube roots are primitive complex cube roots, not real cube roots.
It expresses solutions in forms which may be inappropriate or simply ugly. For example, in the case of cubic equations with
#3#irrational real roots it expresses solutions using cube roots of complex numbers rather than real trigonometric functions. In the case of cubic equations with one real and two complex roots, it tends to express the real solution in a form like #root(3)(a+sqrt(b)) + k/root(3)(a+sqrt(b))#rather than #root(3)(a+sqrt(b)) + root(3)(a-sqrt(b))#
It may provide an instant solution, but does not really describe how you would find it, so does not really help the student to learn how to solve the problem themselves.
Suppose you were given two polynomials and asked to find their GCF. They may have been constructed so that it is easier to manually find their GCF by polynomial division than to manually factor them. However, if you give the polynomials to wolframalpha, it will happily find the factors for you, prejudicing the choice of solution method.
I perceive this sort of thing as a nuisance for teachers who want to give their students good examples to solve.