-snip-
Hence the 'get it right consistently', which includes changing the equations. And I said 'barring using a calculator', as in 'If you use a calculator, you should show your working'
But not enforcing text-based answers causes people to not learn text-based answering. And not learning text-based answering imparts a ton less experience at these problems. And not having experience greatly diminishes your performance on the test. Do you see where I'm going with this? The text-basedness is not there for the talented people, it's there to help the untalented help themselves (and even the talented can still benefit from such experience, if only for the added sense of achievement that you need to transit to effort-based ability before your talent runs out).
I... ah...
DON'T see where you're going with this. Text-basedness... No. I don't see that. I mean, obviously, TEACH the techniques and things, make sure people understand them, and encourage them to use it just in case they screw up, so they can see their mistake, but I don't think they should enforce having to write down the techniques. Again, barring calculator tests.
This here is exactly what's happening. Text-basedness makes sure that the testees know that their understanding is what is being tested, not their ability to enter numbers into a formula. Knowing that their understanding is being tested forces them to understand the material instead of simply memorizing a formula and hoping it's the correct one when they actually use it. It shows them that mathematics is about understanding, not about formulas. And as soon as they properly have the ability to understand, they're basically set for life in maths.
Also, "understanding" is a weird word now.
Oy, just make them write down the intermediate steps! Keeps them on their toes just as well, and avoids the whole sluggishness of language when it comes to mathematics.
(Also, that sum n*b^n thing comes out to something like (b/(1-b))^2, right? To show the intermediate steps:
sum n*b^n= sum (b^n + b^n + ... + b^n) = (sum b^n)(sum b^i) = (b/(1-b))^2