10 Steps to Solve a Problem

This semester, I’m teaching our MATH 229: Vector Calculus with Chemical Applications course for the first time. It’s a 5-credit hour course designed for Chemistry majors who have completed Calculus I. Topics include some pieces of Calculus II, Vector Calculus, Linear Algebra, and Differential Equations. (I’m using a standards-based grading system and I hope to blog about that sometime soon.)

The textbook for the course was written by Jason Howell and although no longer at CofC, he has kindly let us continue to use the book and distribute the PDF to our students for free. Like most math textbooks, each section has some explanation pages and various Examples. We are working through the Examples together in class, and to prepare, I’ve been going through them ahead of time. Here’s an Example from an upcoming section: I started thinking about this problem on January 1st and today I finally produced a solution that made me happy. During the 9-day stretch, I found lots of non-solutions — either methods that I couldn’t get to work, methods that I could get to work but didn’t like, or methods that I realized could work but weren’t suitable to use in my Vector Calculus course. When we reach this problem in class, we will still be in Chapter 1 of the textbook, and my students will know some stuff about three-dimensional space, vectors, spherical and cylindrical coordinate systems, but not a lot of linear algebra or complex variables. I won’t tell you the solution (consider it your homework!). Instead, let’s just consider ways we might find a “suitable coordinate system for the molecule” since that’s really the part I found tricky.

My algorithm was very inefficient as compared to Feynman’s Problem Solving Algorithm, but here it is:

Ten Step Problem Solving Algorithm:

  1. Put one of the hydrogen atoms at the origin, another one along the positive x-axis, and a third somewhere in Quadrant I. Use rectangular coordinates and the Pythagorean Theorem (a lot). Try to find the centriod of the triangle.
  2. Say “Hmmmmm…” aloud often enough that your husband asks what you’re working on, and then do a fantastic sales pitch about how interesting the problem is so that he starts working on it too.
  3. Put the equilateral triangle built out of the three hydrogen atoms on the xy-plane with the origin at the centroid of the triangle, and one of the hydrogen atoms along the positive x-axis. Use what you know about triangles to figure out the distance from the origin to the atom on the x-axis.
  4. Because of input and advice following Step 2, give up on rectangular coordinates and think about using cylindrical coordinates.
  5. Give up on cylindrical coordinates and go back to thinking about rectangular coordinates.
  6. Put the nitrogen atom at the origin and the hydrogen triangle on a plane parallel to the xy-plane, then try to find the distance between the triangle and the origin.
  7. Stop people in hallway and ask for help and input. Convince former students and former graduate teaching assistants the problem is interesting and see what they say.
  8. My office next-door neighbor convinced me that it’s smartest to put the origin at the center of the triangle, so I stuck with that after hearing her arguments about symmetry.
  9. I pitched the problem to another colleague who immediately drew a picture using complex analysis, DeMoivre’s formula, and roots of unity. I had to toss aside this solution since it didn’t follow from the previous material in my Vector Calculus course.
  10. Settle on a solution: Use ij, and k vectors, some vectors of the form aibj, and some known lengths to figure out appropriate constants a and b.

[Another colleague suggested I just get the solutions to the textbook problems from someone else, but (a) I haven’t found anyone who has them and (b) as a matter of pride stubbornness I’ve been doing them on my own.]

I’m not sure how long I spent on this single problem, but an estimate around 4 hours is probably reasonable. I hesitate to mention this since I’m sure the entire internet will leave me a comment of the form, “How can you be so bad at such immediately obvious and simple math??!!” On the other hand, maybe it’s worth mentioning that even those of us who do this kind of thing for a living sometimes find “easy” problems quite challenging and that not being an extremely speedy problem solver doesn’t preclude you from getting a job solving problems.

Also, here are three specific goals I have for myself this semester:

  1. My instincts about problems in vector calculus are not very strong, almost certainly because I have not solved any vector calculus problems since I was an undergraduate. (That statement is probably factually false, but it is a reasonable approximation of reality.) So maybe I can re-awaken those parts of my brain.
  2. I want to get better at drawing things in 3D. I have sometimes wondered if my lack of passion for multi-variable calculus is because I am not happy with my ability to draw the objects? Maybe this course will force me to do more drawing and I’ll get better at it as we go.
  3. I hope to learn some stuff about chemistry — sure, from the textbook and course material — but, more importantly, from my students. I like hearing them talk with each other before class about all the various chem classes they are taking. I haven’t taken a chemistry class since high school (and that one wasn’t didn’t even have a lab associated with it).