Grothendieck is very transparent about it in this passage, which I’ve previously cited: “All my life I’ve been unable to read a mathematical text, however trivial or simple it may be, unless I’m able to give this text a ‘meaning’ in terms of my experience of mathematical things, that is unless the text arouses in me mental images, intuitions that w

That’s why it’s so important when teaching maths to understand the student’s way of thinking, and point out what was wrong with their logic, not just what was wrong with their final answer.