What you hold in your hand or—this is the 21st century—view on screen
Let me explain. Mathematics is often taught by presenting students with a
broad concept, like linear independence, and then a set of tools for identifying,
using, or creating it. More rigorous courses might include proofs, like the
theorem that the eigenvectors for a given eigenvalue form a vector space.
Understandably, this approach leaves most students with the impression that
mathematics is a collection of results and algorithms. But these are merely the
products of mathematics, and are about as interesting to the mathematician as
last week’s bread to the baker or last year’s canvas to the artist. The theorem
has been proved, the bread baked, the portrait painted, and the important
question is: What’s next?
Mathematics itself is the creation of these concepts and algorithms. In a
perfect world, students would begin with a simple concept, and then develop
the tools necessary to handle this concept. This is the idea behind inquiry
based learning (IBL).
The value of IBL should be clear. In the modern world, if an existing
method can solve a problem, then a clever programmer can code that method,
and a computer can implement the method more rapidly, more accurately, and
more cheaply than any human being. So what’s really important is not how to
solve a problem, but how to create the solution to a problem that no one has
solved. Thus, a 21st century mathematics textbook should focus on teaching
students to be creative.