A theme of this blog for several months has been the importance of challenging problems to help students learn. Last week P.J. wrote about one of the most challenging problems every teacher faces: those students who just don't get it, and whom we can't get to no matter what we try. Now that I am retired and am not faced with that particular problem on a day to day basis, I may be able to provide a different perspective.
Of course I agree with PJ: the most important thing is to NEVER give up on the student, no matter how hopeless it seems. Learning is complicated and unpredictable. Some time ago, I was tutoring a student on a weekly basis, one-on-one, for an hour at a time. She was taking seventh-grade math, and it seemed hopeless. I would give her a problem. She would get it wrong. We would go over it. I would give her another. She'd get it. We'd go on to the next topic. Fifteen minutes later, we'd return to the first topic, and she would get it wrong. This pattern happened every session, every week. In eighth grade, a similar experience presented itself as she studied algebra. When high school arrived, she took algebra again, this time at an honors level. She not only passed, but was disappointed when she only got an A-. She managed a B in Geometry Honors and ended up taking four years of math, the last two at a regular level, and going to a good college.
She had resources, and she had encouraging parents who did not give up on her, and she was determined to succeed--but she did succeed, and that is the thing to remember. Each person has his or her own learning personality. Each person can learn, but some in radically different ways, and many are looking for an excuse to quit trying. A teacher who will not give up on the student fights that student's urge to quit.
In her outstanding book, Overcoming Math Anxiety, Shelia Tobias points out that virtually every adult she interviewed had a vivid memory of a moment when a teacher made the interviewee believe that math was not possible for that person to learn. We must ensure that every student believes that we believe that student can learn important mathematics, somehow, sometime. One of my colleagues, Janet Webb, used to continually remind us that every parent sends us the best child they have and expects us to do our best to educate that child.
Another related thought is that the specific content of the course is less important then the intangible things that the student takes away. The impact of their time in your class has more to do with the way you treated them and the respect, enjoyment and excitement about mathematics you demonstrated than the actual algorithms and theorems they were tested on.