Teaching the test?

The above phrase is often meant as a pejorative, though the superior alternative is rarely elaborated upon. One such method from Hungary can be found here: 

https://bsmeducation.com/about/wolves-and-sheep/

The work of Paul Lockhart in this area is also worth reading. His basic thesis is that actual mathematicians rarely care about the practical applications of their ideas, and so the proper way to teach math is to treat it like art or music. That is, it is an expression of creativity with a strong aspect of play.

Common standardized tests like the SAT have a math section with about 60 questions with about 2.5 minutes max per question. Each question is about a different topic, so a well-prepared student will have been taught about each of those topics at some point.

These critical topics include: Pythagorean theorem, quadratic formula, binomial theorem, similar triangles, order of operations, solving simultaneous equations by elimination, etc. There are not that many topics and it would be easy enough for a competent teacher to cover them all in a school year. 

Ideally, students should be presented with unfamiliar problems and given a chance to use their creativity to figure them out on their own. Part of the joy of learning comes from discovery and the aspect of play. 

Instead, what usually happens, at least in most American classrooms, is that the teacher shows the students exactly how to do each problem and then assigns a few dozen nearly identical practice problems. I've heard this method called "drill and kill", and it seems an appropriately dour name. 

Part of the problem is that many math teachers themselves are not particularly good at nor interested in math. Far too many of them see math as formulas and algorithms to be memorized and regurgitated. The problem with that philosophy is that it leads to students who think they are good at math but actually are only good at following instructions. 

Bottom line, "drill and kill" has got to go and the cornerstone of "teaching the test" should be always letting the students try the problem first before the teacher explains the solution. It's also important to remember the Pareto principle in that only a relatively small number of concepts are going to be asked about on standardized tests.