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I've noticed this (backward steps) with some friends. I think it's because people don't think logically with math notation--they just play with symbols until they get the sentence they want.

To get around this, I just get people to argue aloud. People wear their logic hats again when they have to talk to a human and convince them of something.

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Alternately, it's a very useful technique to identify the penultimate step; it's often a useful rephrasing of the problem, and gives something easier to prove. And then just repeat...

(It's probably usually a hangover from poorly taught rote proof exercises in a lower grade, though.)

This is how you do high-school algebra, after all. As long as it is clear that all steps are reversible, it is ok.

Writing the proof in the wrong direction is useful in some kind of problems, for example to prove by induction that 1^3+2^3+3^3+4^3+...+N^3 = N^2 * (N+1)^2 / 4.

My recommendation to my students is to write to "proof" in the wrong direction in a scratch paper, the rewrite it carefully in the correct order in the final paper checking that all the implication go in the right direction, and then destroy the scratch paper.

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