Here are the questions:

What’s he messing up on? Focus in on #2.  How did he get 11 as his answer?

• I see this all the time. Students come in knowing that if there are two negatives signs, you can change them both to plus signs. They very frequently forget that those two negative signs have to be next to each other.

So for these students, the “trick” I give them is to turn the two negatives into a plus, but they’re only allowed to draw one line. If the negatives are next to each other, this is easy “–|–”. If they’re not next to each other, it doesn’t work, so you’re not allowed to change them.

• I think he is messing up with the “back to back” negatives. The concept of minus a negative. I would be interested to give this student follow up problems like: -8 – (-2). I think he would still write 6 instead of -6.

• Two negatives make a positive! Someone just wrote a blog post about this same misconception. Was it you? I’ll have to dig through my reader tomorrow to find it.

• Number 6 is my favorite — the kid has a good working hypothesis about the analogies between addition and subtraction and multiplication and division.

• Mike Jacobs

Two negatives make a positive! I see it all the time and not surprising really because I hear teachers say it all the time (believe me I used to!) For number 6 I’m wondering if the student is twisting the rule ‘the number you’re dividing by, just invert and multiply’? Have they ‘inverted’ the -2 to become +2 and then multiplied? I’d like to give some similar questions to see if this is the case.

• louise

I say that if you have two sticks next to each other, you can pile them up into a plus sign (after we’ve done the two $180^o$ makes $360^o$ bit in several ways). But sticks don’t get to leap over anything, just to snuggle up together. I work with pretzel sticks (signs) and Skittles (items to count ).

• Jinjer Markley

I think there’s more here than meets the eye. The student got #3 correct, so he or she remembers the piling up rule.

How does this student write the number 7? I write mine with a little dash through the middle, which may be what the student mistook for a negative sign here. Why they would get 11 instead of 3, I can’t explain, but maybe the discombobulation of code switching between a dash-as-part-of-the-number and a dash-as-operation tied him or her up.