Monday, October 29, 2012  teaching backwards  One of the most
difficult things about teaching math and science is avoiding the
temptation to teach everything backwards.  The backward approach
roughly follows the following the scheme, and seems to be
standard in every textbook (and far too many classrooms):
introduce an idea, term, or a definition, and then explain what
it means, how it's relevant, and how to do "problems" or answer
"questions" about it.  For example, most math textbooks come to
a chapter titled "trigonometry" or something like that, give
definitions of the trigonometric functions (sine, cosine,
tangent, etc.), and then proceed to show how useful they are in
solving problems involving triangles.  Later in the chapter, in
the section titled "the sum and difference formulas," the book
gives the sum and difference formulas for sine and cosine,
before giving the proofs, and before even stating a relevant
question that would require the sum and difference formulas.  Or
a standard college physics te