My question still holds though; when do you go from thinking "Why didn't they check their work? My students make so many careless mistakes." to thinking "Wow, they really didn't understand this." If a couple students make the same mistake is that a clue that they have a misconception or were they both careless? What if 10 students make the same mistake? On this same test quite a few students used 2π to calculate the period of a stretched tangent function (trig refresher: 2π is the period of sine and cosine, but π is the period of tangent). I'm still not sure if they just forgot in the moment with the pressure of taking a test, or if they don't know that tangent has a different period than sine and cosine. When does an error earn the label misconception?
If you're curious, here's what I was thinking:
And what the students were writing:
And the more precise way to write out the solution to the problem:
I think mine was a mistake because I realized it didn't make sense when I started thinking about it. Is that the difference between a mistake and a misconception?
Seems like that's a really good question. I think it comes down to volume... if one person is making the same mistake multiple times (ie- on multiple questions), they probably don't realize it's a mistake. That's a misconception. Similarly, if a large number of people are making the same mistake once, it's probably a misconception - as it's statistically less likely they're all making such a mistake.
ReplyDeleteIn this case, I'd say it's a mistake because not only did you realize it, you knew what tools were needed to fix it. If you'd realized there was an error, but had no idea where things went wrong, that would imply at least a temporary misconception. Of course, inverses and reciprocals are something of a minefield for both...