So this semester I'm teaching precalculus. Supposedly the students are already reasonably proficient in algebra.
Good parts of the semester include the fact that I found a book that I like. I really, really like it. It's beautifully focused. It feels like every time I start a new exercise, I think, "Yes--this is exactly what I wish my calculus students understood." And it's beautifully structured, spiraling through topics, adding layers of subtlety with each turn. I'm very happy about having such a good book in part because pretty much every semester I have been here so far, I've used books other people picked out or which were "typical" for a course at the university, and I've pretty much universally despised those books. I've stopped trusting anyone's recommendations.
I also like the way I have structured the semester, with lots of "mini" tests, which are cumulative, rather than two or three "big" tests that students cram for. It keeps the students up with the material, and it also keeps me apprised of where my students are. But this means I really do know how my students are doing, and I'm not feeling as happy about this at the moment. After the second quiz, I know there are many things that many of them cannot do. Many fairly simple things that many of them cannot do. Including things we have done repeatedly since the second day of class.
Part of the problem comes from previous deficits. Many of the students have trouble solving simple equations. Several need to be reminded repeatedly that there are real numbers between 2 and 3. Some are not sure what you might get if you were to square the square root of 5, or that -3 < -2, or whether a squared real number might be negative, or whether one might be allowed to take the square root of zero. (It's zero, by the way.) When students are struggling with these issues, it makes it difficult for them to learn about the domain and range of a function, and what the rate of change of a function on an interval might be, and how to sketch a piecewise defined function. How did these students end up in precalculus? Are they really expected to be able to complete calculus next semester?
So I have many failing students now. And tomorrow I must chide them to get the help they need if they wish to pass. There is still time, but the time to catch up is running out rapidly. And for so many of them, there is so much to catch up on.