One of the smartest teachers I ever had was during my final semester of graduate school. She taught linear models in statistics, lecturing without notes the entire course. The course was difficult; getting through each lecture was difficult, and it was the only B I got in grad school, and I counted myself fortunate I got that. While certainly brilliant, she expected her students would comprehend the material the way she did. I went to her office for help not long after the semester started and left feeling this was going to be a difficult course, because I still didn’t understand. In my view, she expected me to, and I didn’t. My fault? Sure, at least in part. But I needed another approach to understand the material; one approach I did do, with some success, was to get many more problems to solve. Unfortunately, I could only find a few such problems in the library, back when we went to libraries.
Note: libraries had other uses, too. After I broke my clavicle in a bike accident, I noted several weeks later that I had trouble shelving a book with my arm. Realizing that the therapist’s advice to exercise the arm might not be so helpful, that night I took a jar of jam and found I could not lift it all the way up from my side. I diagnosed a suprascapular nerve injury from the fracture, affecting my supraspinatus and infraspinatus muscles, which would improve only with time, not exercise. It took six months.
There were brief glimmers of understanding during the course: when I finally figured something out one Sunday afternoon on campus, the teacher happened to be present. I was so excited that I told her what I had figured out.
“Well, that’s straightforward,” she replied. “Nothing to it.”
I was deflated.
I have taught math in a variety of places, to young and old, motivated and not, ex-cons, people who couldn’t tell time on an analog clock and if one approach didn’t work, I had others. I seldom said something is easy. When people are struggling to keep their heads above water, I didn’t tell them that swimming is natural.
My thesis advisor had a gift of telling me exactly what I needed to know and nothing more. If I applied myself after talking to him, I would solve the problem. And I always did. I swore a few times doing it, said he was jerking me around, but he made me think how to solve it and knew I could. That is a great teacher. I was never spoonfed, but I wasn’t left out on a limb, either, staring at the leaves, wondering when I was going fall with them.
Once, while writing my thesis, I went up to his office to tell him I had figured out the first two moments (average and variance, the ones that matter most) of a function the two of us were studying. I showed him the approach I used. We discussed it for awhile, and when I left, he said, “Nice proof.”
Wow, he didn’t hand out a lot of compliments, but when he did, they mattered. That’s another sign of a good teacher, too, to encourage. It was a slick proof.
I’m two decades away from both of those professors now. Retired, I go on a math website and try to solve problems that people send from all over the world. Of the 11,500 problems I have solved, about a quarter of the students have sent me comments, and over 95% of those are positive. Not all the negative ones are my fault, but certainly many are. Yes, I have made mistakes. We all do, so I try to choose problems where I can prove my answer.
Recently, I got a thank you note recently from one student. I checked the problem to see what it was. I get some of the nicest notes from solving basic arithmetic problems, but this was a sequence problem, and I got it wrong. I knew that, because at the bottom, another tutor wrote down the answer—no work—adding, “Another one who gets a lot of wrong answers.”
That’s a huge downer, especially since I have no idea how many of my answers are wrong. I decided to learn more about quadratic sequences, and I solved the problem along with one that the student asked me to in his note. I didn’t check to see if the other tutor had added work. I doubted he did.
It’s rude to make nasty comments online, but rudeness has been the mainstay of the country for several years now. I have found wrong answers from others, too. I write down my work and let the student see it. I don’t want to get into arguments with people while I am trying to help others. Writing down only the answer on the site will help a student who just wants the answer and to move on, but the purpose of the site is for students to see how to solve a certain kind of problem, and the man who criticized me didn’t do any of that. I apologized online and wrote the answer clearly.
I decided to look at one of the other tutor’s problem solving lists. I had seen some of her comments before in passing, and they were often belittling. She had a comment about one of my problems, but I never saw it because the student hadn’t acknowledged my work. She said that I had done “tons of extra calculations,” that there were better ways to solve it, and I needed to make sure I didn’t hurt “my reputation.” Well, there were better ways, but I picked one that worked for me. As for my reputation, nobody there knows who I am. Remember, this is the Internet, and if people were known by their name and address before they could comment, a lot of silicon chips would return to sand, unused.
I then looked through her solved problem list. I didn’t appear again, but she took issue with students who submitted more than one problem (one of whose multi-submission I subsequently took care of). She complained about their English, when many of the students were non-natives and wrote what they could. One student didn’t write the problem clearly, and rather than trying to solve what she thought he might be trying to say (we can do that), asked him to stop submitting.
A good many problems on her list of problems “solved” are those where she wrote snide comments and didn’t answer the question. Mind you, she knows her stuff. But she tends to be very picky about with whom she will share that knowledge. She may be smart, and she may know how to teach, but she will never be a great teacher, and she may not even be a good one.
Instead, she is a bully. She beats up on those who can’t defend themselves, which I learned as a boy one does not ever, ever do. It is rude, inexcusable, and unfortunate. And it is how America works these days. Look only at the statehouse rallies recently.
I think, although I can’t remember for sure, she once gave an answer that while correct, wasn’t what the problem asked for. I have seen mistakes and just do my work, not try to correct others. I tutor in person at the community college, and occasionally I am asked by some to deal with certain problems. I have asked for help, too. We all do, and nobody keeps score. We are there to help students. Online instruction can be great, but again, online anonymity is the worst aspect of the internet. It allows for all the unfiltered anger, hate, and stupidity to appear.
Making snide comments in public is especially devastating. Helping in a classroom 10 years ago, I saw where a teacher was about to make an error in solving an absolute value equation. He was going to get stuck, and it would take a while for him to figure out what to do. In the meantime, the students would become restless.
I walked up to the front of the room, quietly took the teacher aside, and wrote down how he needed to address the problem. I mentioned that it was easy to get fouled up at this point, because I have. He took it from there, finished the class, and the students never knew what happened.
I am picking and choosing my problems on the site now, which is too bad. Right now, I don’t want any more nasty comments about me until I have done at least a thousand more clearly proved examples. My apologies to those students whose questions are on the border of what I can do quickly and easily. Rudeness and bullying harm so much in this world.
Boreal B[l]og 
Leave a Reply