One of those "teaching is such a grind" moments. I don't know how common this is, but I'm constantly thinking about my courses in the long term and super immediate short term. The march toward May constantly at odds with what we're able to accomplish in any given 50 minutes. Generally speaking I think I know how to help students learn something, but what gets internalized and how students make personal breakthroughs is just such a mystery.
Compounding the problem is an issue I think teachers fail to recognize sometimes, the ever growing gap between my fluency in a subject, and where students are when they get to me:
The longer you teach a subject, the greater your comfort level. At a certain point, you can recite the entire curriculum without reference, and walk someone through the connections to past and future courses. You can Calculus in your sleep.
Students don't have the benefit of running through Calculus over and over and over again. They're seeing it for the first time, expecting you to be their faithful guide. Their base fluency in math will be about the same year after year. And that's where it can become frustrating. You forget what it's like to learn derivatives for the first time, and what do you mean I have to explain integration again, don't you get it yet!?
I've never ranted out loud, but we've all been there, explaining something that just feels so simple for the 5th time to someone who isn't there yet. Constantly reminding myself of the fluency gap keeps me sane. You can't teach 15 years of math grind in 8 months.
And yet, the struggle remains real. Calculus has moments where we make immense progress, and then I set them loose on an assignment and I seem to be answering very basic questions over and over again. I have options. I can push forward because darn it we have a curriculum to get through, get mad and rant about their dedication, or find ways to give them time. In most cases the students are asking the questions from a genuine place. They really want to make the connection, they know the explanation was good but gosh darn it Mister why isn't this making sense to me?
I've run through the AP ringer twice now. I know the simple things are the enemy. It's not worth bothering with the complex scenarios and phrasing if they can't do the simple things. What good is recognizing an integral is necessary if you have no idea how to integrate?
So we stop. I give them a small mountain of derivatives and integrals to chew on for a week. They get better, and we slowly start moving forward again. Despite having to take a couple of pauses, we're still on track pretty well. But there's a balance here. You can't do this all the time. Kids need a push. It's very easy for them to say "we're good" and bring the pace to a crawl.
The data seems to indicate this is mutually beneficial. We've done two benchmarks in Calculus so far, and the students are doing about 15% better year over year. But it's easy to get excited about that data. Sure, averages are up, but what does that mean? Last year I felt super confident with my data, and got burned. I now read the numbers with a more skeptical eye. If only 10% of my students scraped together a decent performance last year, what does 15% better mean? At the same time, I know I've pushed this group harder and placed more emphasis on vocabulary. So what does 15% better on harder material mean? I have no idea.
The point is to say that it's very easy to obsess over the end goal, whether that's a district-level or campus-level goal, state testing or national testing watermark. Those end goals are good external motivators. But the struggle will remain. You've still got inexperienced learners who are going to take exactly the amount of time they need before they start improving their fluency. Kids just take forever sometimes, it's what they do. There are tools to speed this up, there are more efficient ways to teach a subject the next year, but if they just need a few days to work things out, are you willing to give it to them?