Finding errors by asking how your answer is wrong

One of the most common situations we face in the MLC is when a student says, "I'm wrong, but I don't know why". They've done a fairly long calculation and put their answer into MapleTA, only to get the dreaded red cross, and they have no idea why it's wrong and how to fix it. One of the major problems is that many students can't tell if it's because they've entered the syntax wrong, or done something wrong in their algebra, or completely misinterpreted the question, or if MapleTA itself has a bug and isn't accepting the correct answer.

The other day, I was helping an Engineering Maths IIA student in exactly this situation. He was solving a differential equation and his answer was wrong, but he didn't know why. As usual in this situation, I encouraged him to think of a way they could check his answer for himself (I commented on this a few years ago, actually Who tells you if you're correct?). In this case, subbing the solution back into the original equation is a useful approach.

When he subbed his solution into the left-hand part of the equation, he got a result of -3/16 cos(1/4 t). Unfortunately, the right-hand part of the equation was -3 cos(1/4 t). So yes, his solution really was wrong. This left us with the much more difficult question of how to fix the error.

In a sudden flash of inspiration, I realised that the way that his solution was wrong might tell us something about the kind of error he had made. How could he have gotten -3/16 cos(1/4 t) instead of -3 cos(1/4 t) when he subbed into the equation? Perhaps because his solution was 1/16 of what it should be. I went looking for a 1/16 but couldn't find one. Ok then, how could you produce a 1/16 in a less direct way? Perhaps you could divide by 4 twice. So this time I went back through his working looking for 4s. Like a moth I was attracted to this line in his working: "A/4 + B/4 = -3 ⇒ A + B = -3/4". Of course! Dividing by 4 instead of multiplying by 4 would have the same effect as dividing by 4 twice, which could totally have produced that 1/16.

I was floored by the amazing effectiveness of this approach, and I wondered that I had never thought to do it before. It seems like such an obvious way to come up with something specific to look for. Admittedly it might not always yield useful results, but the evidence from this episode suggests that it might, which is certainly better than no strategy at all!

The student himself was suitably impressed and you could see him consciously committing the idea to memory for future reference. So now at least two people have a new strategy to find errors: when you sub your answer in to the original and it doesn't work, investigate the way that your answer is wrong – it might help you find something specific to look for to find your error.