Yeah, I'm personally of the opinion that if you CAN get it right consistently, you shouldn't need to show your working out, barring using a calculator (for obvious reasons).
Apparently, though, that's not enough.
Using calculators to solve problems for you ensures that you're pretty much boned when the type of problem changes ever so slightly. Imagine the problem "x²+3*x=4, solve for x". Just whip out your calculator, input the numbers into your quadratic equation solver, get two results, done. Now solve the problem x²+a*x=4. Your quadratic equation solver can't handle the a? You're fucked, congrats. You actually need to know the quadratic formula (or at least know how to derive it) to solve this problem. Same thing applies to every type of problem you get in high school. Take a random problem you can solve with a calculator. Now replace one of the numbers with an a. Suddenly your calculator is a useless pile of trash.
Now doing it in your head has the same type of inherent danger, albeit with a higher threshold. Imagine a problem requiring you to sum b
n for all positive integers n. If you know a formula for that, you can simply insert the formula and do that in your head, the answer is b/(1-b). Now imagine a problem requiring the sum of all n*b
n. You can't work that out via a formula since you've never seen such a series before, but you could in fact write that as an infinite sum of infinite sums and get b/(1-b)² as a result. Now my point is that the confidence in trying such counterintuitive approaches is not at all dependent on talent, it depends solely on the amount of experience you have with similar situations, and you get much, much more experience from a problem if you take the time and write down the exact thought process leading to the solution (and if the test requires written solutions, the students will practice written solutions, thereby gaining such experience). Talking from my own experience here, and trust me, I have a lot of that.