Upon the request of the OP I'll elaborate on my comment:
If the AP and other test makers were doing their job the test would be vastly different from year to year precisely to discourage teaching to the test.
I have a variety of different data which informs my disdain for the AP test as it is currently practiced as well as the "SOL's" (Standard Of Learning) tests which shape elementary math teaching in Virginia (where I used to live).
I'll begin with the SOL's. I had a neighbor who taught math to middle school kids, he described how he taught them "tricks" to get the problem right on the SOL. From what I heard, the focus of the learning was on the tricks to help pass the SOLs rather than on building a unified mathematical framework infused with both concept and calculation.
With the AP test, I think they're allowed a graphing calculator now ? Yep:
So, that's not a good sign. I can't see a reason you would need a graphing calculator for a well-designed calculus test. Unless you're testing ability to use a graphing calculator (why on earth encourage that ? there are far better alternatives online and... again, really inappropriate to a calculus placement test). So, I don't have a lot of firsthand testimony from students who took AP calculus and described their course, but I know that as a department we saw reason to be less and less trusting of anything but a 4 or 5 on the exam. At that point, it's more an indication of intelligence and study habit than actual knowledge of calculus.
Why ? Because the test is largely unchanged, so if you study the standard problem types you'll be able to make a good score if you are intelligent. Your inability to add fractions etc. can even be hidden by the ever present calculator.
Or, if you're really clever and a cheat, you can program the answers of the test into your graphing calculator after a scout student takes the test and publishes the questions and answers on the appropriate "help" site. Here I conjecture, but anyone who has experience with the Chegg homework feedback loop knows my speculation is not an idle one.
Beyond the specific situations I list above, I think the general response we receive in teaching students speaks to the abuse of math in schools. I have had countless students (good students) come to me mid-semester the first time they have my course and say something along the lines of "your teaching is so different". Why ? Well, I think there are a couple things I do which are never going to be high on the priority list of a teacher who teaches to the test.
- emphasis is on definitions
- proofs of theorems which involve the methods of the course are given
- compare and contrast definitions
- ask inverse questions
- contrast graphical verses analytical methods
- warn about pathological cases
- discuss historical evolution of math concepts
In contrast, a person who teaches to the test has two main questions:
- which questions are allowed on the test ?
- how can we reliably do the questions as quick as possible ?
In the above mode of teaching, definitions may not even be stated, theorems are seldom proved and the bulk of time is spent solving problems. I am guilty of such teaching in my absolute lowest level teaching, but to teach calculus this way is to rob students of beauty.
This danger is not isolated to calculus. In fact, any suitably standardized test or curriculum has the danger of squelching creativity and the best teaching for the sake of the pragmatism of passing the test.
But, the existence of a standard test is not the automatic death of teaching. For example, my wife took advanced math in highschool in Hong Kong. They have to take a test which is quite serious just to get into the math track if I understand correctly. That test didn't discourage good teaching, rather it promoted it because the test was a bit unpredictable and their were many different kinds of problems you might face. That test was administered in such a way that the teachers who taught the students were not the teachers who proctored the test (a much more serious business than what we find at American institutions).
I'm torn on the merit of standardized testing. In principle, I see great merit in replacing course credits with passing certification tests because many students are trapped in universities whose standards are so low that professors are held hostage by the weakest students.( I can envision replacing a math degree with holding certifications in different subjects. That would free students from the burden of liberal arts dogma) In practice, we cannot teach the course as it ought to be because, well, the students. The existence of an external test which created without direct regard to the weakness of students at a particular institution gives serious students a way to prove their skill. That said, such a test once it becomes too familiar after years of creation and the perhaps inevitable recycling of questions, becomes gamified and discourages good teaching of the topic.
This is why I made my comment, in my view the creators of standardized tests must engage in a certain randomness to fight against the gamification.