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.)

If you have memorized the general heuristic to solve any particular equation, its basically that is all there is to solving that equation. What is left after that is to practice that algorithm enough number of times until it just comes to you naturally when you see such equation again.

Richard Feynman, noted people find math difficult because all the while they are focusing on just that 'memorizing that algorithm' part. He used to call this as 'for people who don't know what they are doing'.

In school, or even in college, there are only a finite number of such algorithms to rote memorize. If you do, and practice those problem enough number of time, you can essentially score big marks in the exam.

Unfortunately the same is in software interviews these days. The most common algorithm experts you come across these days, are exactly algorithm experts. These are simply people who can recall all the algorithms they have rote memorized earlier. I know of people who spend dedicated time every day memorizing time and space complexity of many commonly asked interview algorithms on flash cards.

Ironically hiring such candidates at times is almost synonymous with hiring bad candidates.

This is actually possible, if you've been taught to solve problems instead of being taught to apply magic formulas. Of course, one is rarely taught math this way now.

Let me try. Suppose you have 3x^2+5+2x=7. Now, you have no idea how to solve for x. But what you could solve? If you had something like (ax+b)^2 = c, then probably this would be easy for you to solve, right (of course, provided c is not negative, won't get into that now)? We just take square root of both parts, and then it's a simple linear equation. Now, we can notice we can also get rid of a, for simplicity, by dividing both parts by a and get something like (x+b)^2 = c. Which in our example should represent 3x^2+5+2x=7 or alternatively x^2 + 2/3x - 2/3 = 0. But how we find proper b and c?

So, is there a way to make that equation look like that with suitable values? Let's look at (x+b)^2 = x^2 + 2bx + b^2. We have 2bx = 2/3x which means b = 1/3. So we have (x+1/3)^2 on one side, and on the other side we have to get 2/3 and also additional 1/9 which is the b^2, meaning c would be 7/9. So, x + 1/3 is sqrt(7/9) with + or - sign (since square on non-zero root always has 2 options), and x would be that minus 1/3.

As you can see, we used no special magic formula - only thing we needed is how to write (x+b)^2, which is easy to see just by multiplying (x+b)*(x+b) by hand.

You can do cubic equations this way too, but it is considerably more laborious. Look up Cardano and Vieta methods. Doable but I wouldn't enjoy doing it by hand too much.

Remember, the goal of an exam is to measure if you have the knowledge. Whether you got the knowledge by studying hard in a traditional classroom setting as scheduled by a syllabus is irrelevant.

Of course, I do not see it as "gaming" - I see it as learning the right way. IMHO, all learning should start with that and formula should come after - as a shortcut, which you can take but you know if needed you could do it from scratch because you understand where it comes from.