The backward approach roughly follows the same standard scheme, as exemplified in every textbook ever (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" using it.

For example, most high school 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.

Unfortunately, this approach completely eliminates any and all creative insight into the very real problems at hand, and leaves the student with no reason or desire to acquire that insight. It's as if the definitions of the trigonometric functions were handed down by God, followed by a set of problems to solve that require them.

In the real world, of course, everything happens in the

*opposite order*: definitions don't lead to problems. Problems and questions lead to insights, which lead to generalizations, which eventually lead to generalizations and definitions. Someone tries to find the distance between two points knowing their respective distances to a third point and one of the angles. After studying geometry (or even before), most students have an intuitive sense of how to approach this problem, and with some care and coaxing, you can get them to "discover" the law of sines. But if they learn trigonometry from a standard textbook, they'll never even get the chance, because there it is before the problem is given!

Most students can learn to use and manipulate trigonometric functions fine by the backwards method, but its flaws aren't merely aesthetic. If a student has no sense for the scope of a problem he or she is trying to solve, what reason does he or she have to remember the trigonometric functions beyond the next test, or the SATs? For most students, the sum and difference formulas are something to memorize and then forget, rather than a beautiful solution to a seemingly intractable problem.

On the first day of my 4th grade science class, I handed the students a sheet with just three questions: What is science?, Why do we care about/study science?, and How do we study science? Their answers were revealing and kind of depressing (also somewhat hilarious). How do we study science? Why, from science books of course! Why do we study science? So that if we need something to fall back on, we can be science teachers!

These students, like so many, have completely missed the point (so far at least!). Maybe some of them will miss the point either way. But it seems more likely that the things they learn will leave enduring memories if they have to confront the same problems that the people who actually discovered and developed them had to confront. At least that way, they get some sense of what math and science are really about!