One of my students came up (with some help) this procedure for converting between degrees and radians.

  1. Memorize the fact that 60° is π/3 and that 30° is π/6.
  2. Note that 10° is therefore π/18 and that similarly 1° is π/180.
  3. You can then take any degree measure and convert it by converting the number of degrees into sums of degrees where you know the conversions. For example, 70° is equal to 60° + 10° = π/3 + π/18 = 6π/18 + π/18 = 7π/18.

Obviously this procedure is not by any means the most efficient way to convert between radians and degrees. Although I showed a much more efficient algorithm for converting between degrees and radians, it didn’t make sense for this student, and so he and I came up with this procedure (which I drew out of him by asking him questions about the angles), which he does understand.

In general, I’d prefer students use inefficient techniques that they understand completely than highly efficient techniques that they do not understand. Hopefully this student will continue to work on his procedure to make it more efficient as he has to use it over and over again, but if not, at least he will be thinking with something that makes more sense in his head.