I have three Algebra I classes this semester. Here is a problem I gave them on a test:
Out of 105 students, only one got it right.
Some students did order of operations wrong (3-3 first, getting 0 as an answer) but a good number were saying 3-9 is positive 6.
Informal checking later led to many students (but less than the test) getting this wrong
and even less getting this wrong
Two things seem to be at work here:
1.) When the problem is difficult to do “in one’s head” and requires some pause, the integer rules are properly applied. However, when it the answer is “obvious” the associativity of the brain triggers the instinctual answer, bypassing any rules (9-3 = 6, so 3-9=6).
2.) The seems to cause extra-super confusion in that there’s an added indirection. Many students who knew the order of operations quite well were answering zero because the associative part of the brain put an override (3-3 instantly shouting out “zero”) and their rationally thinking got blasted. If they managed to resist their mental instinct and get past that successfully they had even less resistance to prevent jumping to answer of 6 in the 3-9 statement.
I suspect even some strong advanced students would get the first problem wrong; it’s almost like a trick question even though there’s none of the traditional hallmarks of one.