Here is a common process I witness in office hours:
- Student struggles with a difficult concept.
- Student gains understanding of the concept.
- Student, now bewildered since the concept is mastered, declares "Oh! That's it??."
They not only learned the concept, but they have declared the concept to be easy!!
The student then maintains the following incorrect perception: "I only understand easy math."
I hear this all the time, at every level.
- Some percentage of students in Calculus II think that Calc I was easy, which is why they got an A, but now Calc II is hard, so they are not succeeding.
- Some percentage of students in algebra think that arithmetic was easy, which is why they got an A, but now algebra is hard.
- To be honest, even I personally have the perception that everything before Hartshorne's Algebraic Geometry is relatively easier, which is why I did well in it!
How many of these people are correct? Is there any research or profound wisdom on this issue out there? I'd prefer research but I'm not sure how it would even be conducted.
- All math is approximately the same difficulty, from arithmetic to algebraic geometry. People slip up because of a bad semester somewhere along the way and that is why anyone ever gets stuck in math, or
- There are "hard spots" in math where you can get stuck; the hard spots become more and more dense as you progress further in math, or
- Something else??