Standardized tests are almost always multiple choice. So, given four (or five) possible answers for x, plug them all in and see which one equals 7. That's arguably faster than using the quadratic formula to solve the problem.
Both of those seem unusual to me and I don't see how that method could possibly be beneficial in assessing knowledge.
Also there will be a lot of "stupid" answers on MC tests like listing three, four, five zeros for a quadratic to see if you understand the basic concept of what you're doing, always read the answers before you start. If you think you're getting five zeros out of a quadratic by some unusual plug and chug of the formula then you've already failed.
Another fun one... the discriminant of this one will be 28, right? Its simpler than solving the whole problem, I did this in my head in a couple seconds even though I'm pretty much asleep. So you'll have a bunch of boring rational number zeros which can't possibly be correct for that discriminant and one whacked out obviously irrational answer and that will be the correct answer.
This also only seems to be an issue for the narrow set of problems where checking the answer is a lot cheaper than solving it. Sounds to me like standardized tests are on the right track, we may just need a few small tweaks.
The first thing any teacher will tell you about UK math tests is write all your workings, because these are marked by a human. So if you're solving some complex statistical thing, and you type 0.535346 instead of 0.545346 so get the wrong answer, but all the steps are right, you get almost all the marks. There are so many places in late high school math where this matters that I can't imagine any use of a test that doesn't have a human look over workings.
Given there's probably only a few questions on each assessed topic, it seems much more important to know whether the student can apply the steps for solving a problem than to just give no marks to someone who makes a minor data entry mistake under exam pressure.
I thought math testing was already pretty bad because revision can be optimised so easily by learning to apply steps (not derive them) and by just doing past papers - but I had no idea anybody actually used an all or nothing approach to assessing these questions.
I don't even understand why you would mark it like that, unless you're only assessing math and other subjects you can feasibly reduce to objective answers. I mean, the point is presumably so a computer can mark it? But if you still bother assessing, say, English or History, presumably you still have to have a human mark them essays?
The goal is to have a score which is a) standardized and objective b) well correlated with mathematical knowledge in the curriculum.
Bsilvereagle identified a flaw with my particular example - students can substitute an easier problem for the real problem. But there are mechanisms to fix that. So given fixes for particular flawed questions like that, it seems like a typical standardized test will provide both (a) and (b).
It would be quite a coincidence if a student could write down the answer to many questions without knowing the intermediate steps, no? (I'm of course assuming security is sufficiently strong, i.e. no cheating.)