The first one looks like summing two different correct methods: there’s the 5 + 7i and the -2 and the -3i in there like there would be from turning the subtraction into addition of the opposite, but then there’s also the 3 and 4i that come from subtracting the corresponding terms. Pick one, and this student would be fine. I don’t know how deep the confusion runs that lets the student do both and put the terms in such a jumbled order, though.

The second one is simple: our horrible notation of putting things next to each other to indicate multiplication, combined with our bad notation of using the same symbol for subtraction and for taking the opposite, has led to this student confusing “things minus stuff” with “things times (opposite of stuff)”. I would apologize to this student for our history of bad notation and talk about how to distinguish subtraction from multiplication.

The student from the first one was just as confused as I was when we discussed that question. Has anyone successfully taught students to check their work?

I got some more insight into the first one: it’s FOIL! Subtract the first terms, and you have 3. Then the outside, which is 5 – 3i. Then the inside, which is 7i – 2. And finally the last, which is 4i. Ha! FOILed again. I wish we could never, ever, ever teach that acronym and shortcut and just stick to “distribute”.

I have had some success in getting students to check their work. I had a rubric (which I can’t seem to find) where they were graded explicitly on things like giving a reasonable estimate with a sentence of explanation, and another one where something like 25% of their grade was based on their choice of writing a sentence or two convincing me that their answer was reasonable or explaining why they thought their answer didn’t make sense.

Cutting down the number of problems I would give on a quiz or exam also helped a lot.