My new favorite game is trying to classify math mistakes. (See: Classifying Math Mistakes)
Right now, I see three big categories of mistakes:
- Mistakes Due To Limited Applicability of Models
- Mistakes Due To Applying Properties of a Familiar Model in an Less Familiar Situation
- Mistakes Due to Quickly Associating Something In Place Of Another
I think this is pretty clearly an example of the third category. The student’s brain was working hard, and they swapped the 10 and the x.
These sorts of mistakes are interesting to me because I think a lot of teachers see these and say, “Oy, this student thinks that you can just swap out the x with the angle.” Or others would say, “Oy, this student has no conceptual understanding of trigonometry.”
Nah. This kid needs more practice with the Law of Sines so that you’ve got enough brain power available to pay attention to all the moving parts while you’re trying to solve the problem.
There’s something else that’s interesting about these associational errors, and it’s about the associations that students make. Isn’t it interesting that the x*sin(10) is more familiar to this student than 10*sin(x)? Maybe this also points to the need for more practice that mixes up missing angle and missing sides Law of Sines problems?
Why did this student think that this verified the identity?
A reflection from the submitter:
I think this 10th grader is saying .174>.34>.5. I wonder what she would have concluded if she’d followed the directions and rounded to 3 decimal places? Many kids were tripped up by the .5, maybe she’d say they were increasing except the .5?
Do you agree with the submitter’s assessment? How do you help a student learning trigonometry nail this down?
I think it’s important to say something more subtle than “this kid doesn’t understand decimals.” One thing that this site has documented is that kids can understand something at 1:00 and then do something entirely different at 1:01. It’s best to see this not as a failure of decimal knowledge, but maybe a failure to use decimal knowledge in this situation. (Some people would say this kid’s knowledge of decimals in a certain context failed to transfer to this problem.) The difference is in how we respond. This kid probably doesn’t need the “basics” of decimals. We just need to make a connection to somewhere where she knows about decimals, I’m speculating.
The submitter asks: What would you write as feedback here? What would actually help this kid?
I ask: What’s the role of written feedback, more generally?
(Thanks again, Tina!)
Hypothesis: Proving trig identities unteaches functions.
After all, thinking of these as functions really just gets in the way, so all of our sensible students just treat these functions like variables.
What’s up with this cool work?
What’s going on here? Tell a story of how this kid wrote this stuff?
Thanks to Jonathan Newman for the submission!
What does this response reveal about this student’s understanding of the basic trig functions?
(Bonus! What’s up with that response for distance? Can you explain why that mistake is a tempting one to make?)