### Rules of Exponents

You might remember that I have this freakishly competent calculus class. They come to class; most of them do the assigned work; they earn high scores on the assessments. For the first time ever, I am teaching a non-honors course where the grade distribution is skewed towards the high end of the scale. Some of my students are telling me that they really like what we're doing now -- infinite series. Definitely an anomolous bunch.

Sometimes I forget that they don't know everything that they should have learned in Calc 1. Looks like on Monday I'm going to have to devote half the class to limit review. On the homework they reasoned that limit as n -> infty of sin(n) / sqrt(n) doesn't exist because the sine oscillates. Really should have anticipated that. Even good calculus students can be shaky at limits.

When I feel a real shock, though is when I forget that they are products of the same public school systems that give me the weak and struggling students who usually populate my classes. Over the past few days several students have come to my office to ask me questions about computations that I did in class. We'd been working with the geometric series, and we've been applying algebraic manipulations to the general term so that it's of the form

Mass confusion. Totally lost. My very successful and accomplished calculus students were unable to follow the algebra. They couldn't see why those two expressions were equal. Working through the problem, slowly, in my office they would ask, "When you multiply, do you add the exponents?" Another student asked, "Is 3

This semester I'm lucky that most of my students have a fairly robust knowledge of algebra with only a few gaps -- and the committed work ethic necessary to be successful in a course that they are not quite ideally prepared for. Just imagine how well they'd be doing if they'd gone to better public schools.

Sometimes I forget that they don't know everything that they should have learned in Calc 1. Looks like on Monday I'm going to have to devote half the class to limit review. On the homework they reasoned that limit as n -> infty of sin(n) / sqrt(n) doesn't exist because the sine oscillates. Really should have anticipated that. Even good calculus students can be shaky at limits.

When I feel a real shock, though is when I forget that they are products of the same public school systems that give me the weak and struggling students who usually populate my classes. Over the past few days several students have come to my office to ask me questions about computations that I did in class. We'd been working with the geometric series, and we've been applying algebraic manipulations to the general term so that it's of the form

*ar*^{n-1}(our indexing starts with*n*=1). We had a problem where the general term is of the form 3^{n}/4^{n+1}. To get it to match the form in the book, I used rules of exponents to rewrite it at 3/16 (3/4)^{n-1}.Mass confusion. Totally lost. My very successful and accomplished calculus students were unable to follow the algebra. They couldn't see why those two expressions were equal. Working through the problem, slowly, in my office they would ask, "When you multiply, do you add the exponents?" Another student asked, "Is 3

^{n}/4^{n}the same as (3/4)^{n}?"This semester I'm lucky that most of my students have a fairly robust knowledge of algebra with only a few gaps -- and the committed work ethic necessary to be successful in a course that they are not quite ideally prepared for. Just imagine how well they'd be doing if they'd gone to better public schools.