In the early 20th century, a student studying mathematical analysis (calculus) in college would probably sit with paper and ink, computing a few dozen integrals. In the middle to late 20th century, mathematical software became available that was able to compute these integrals, and in the late 20th and early 21st century, this functionality became available to students in the form of TI-89 calculators. Still, though, high school teachers and college professors almost always required the students to "show their work", rendering the automated integration technique useful only as a way to check the answer.
This changed very recently with the release of Wolfram alpha. Here is an example of it doing integration for you, with both words and equations. One needs only to copy the text into LaTeX and turn it in.
Impressive, huh?
Actually, while the functionality of the Wolfram Alpha search engine is indeed impressive, this is not the impressive part. Here's why. Software uses algorithms for numerical approximations when numerical computations are required, but algebraic manipulation is pattern matching, just like it is done on paper. To a first approximation, a computer solves an integration program just like we do. These algebraic computation systems have been developed for a long time, and the algorithms are accessible in full to someone who is willing to spend some time on it. All that is left is to print the steps.
To calculus teachers reading this: please don't deny students access to this resource. Just don't allow computers on exams.
1 comment:
Actually, the TI Nspire CAS shows all its work as well and was out long before Wolfram Alpha. And in addition to doing calculus, it can do linear algebra.
Post a Comment