I once got a proper shouting at school. I have an uncanny knack with maths that I can 'see' the solution in my mind and would often just write the answer down. The teacher used to berate me all the time for not showing the working out. one day I submitted my homework with pictures of a head with gears whirring inside or with fingers pressing buttons on a calculator. He went absoutely ballistic!
We wre allowed calculators in my day(1985) but you culd equally do it all in your head. In fat calculators often made it harder as tehy naturally 'rounded' teh answer and when you have calculations that are 15 - 20 'layers' deep these rounding errors would accumulate to give teh wrong answer.
e.g, sin 45 = 1/square root of 2 ( hard to write on a post but always what i would use ) or 0.7071068 to 8 significant figures. use teh latter as a basis of a calculation and you have 'gained' a small fraction that will mean your next layer will be that bit further out. continue and its soon pushing you further and further away from teh 'correct' answer.