Posted in Assessment, Calculus, First day

AP Calculus BC: reflection and ideas for the future

I loved teaching AP Calculus BC last year! I was a little intimidated at first… I hadn’t taught or even used Calculus in 15 ish years. It was a great experience and I can’t wait to do it again! But alas… not enough sections this upcoming year. I do want to reflect on what I did and what I would like to do in the future so I don’t forget.

My favorite lesson was actually the first day of school. I showed three linear graphs and asked students to write down everything they knew about them.

First day graphs

After talking about key features – intercepts, end behavior, domain and range – and linear vs. constant functions and a line that’s not even a function, I surprised them: these aren’t lines!

First day graphs 2

Students worked in teams to match each graph with an equation and come up with a convincing reason why. Then teams presented their solutions via gallery walk.    

First day student 2   img_1357

We touched on domain and range, even and odd functions, and other key features of functions; we also began exploring the ideas of limits and local linearity. Students really latched onto the idea of “zooming in” on the graph, which ultimately paid huge dividends throughout the course. I found myself saying weekly, if not daily, “remember the first day of school when we zoomed in on those three graphs?”

Another win last year was practicing derivatives and (some) antiderivatives at the same time. This is one of my favorite teaching strategies in general; if you can do something one way, you should be able to think “backwards.” I only did basics, like the table below, but it really got students thinking both directions.

I like to give a little peek into what’s coming, and since we already knew that the derivative of a constant is zero, it wasn’t too challenging to sneak in conversation around this “+C” business early.

Here’s another one, this time after trig derivatives and chain rule:

In the future I would like to do this even more intentionally and often.

Around the same time, we were sketching derivatives given a graph, so I threw in some antiderivatives in the same way: if this is the derivative graph, what could the original graph look like? Here they could really see the “+C” as teams came up with different vertical translations of the same graph.

As much as I loved exploring and wondering about these different ideas, the biggest struggle I had was the order in which to introduce concepts. Everything is so connected, and I didn’t want to muddle their understanding by doing too much at once. I ended up sticking pretty close to our textbook; especially in my first year, especially since my PLC teammate would also be following the textbook, especially since the instructor of my APSI used the same textbook at his school and provided me with a bunch of already aligned supplemental resources (!!!), it felt important to stay consistent.

Now that I have a little more “big picture” vision, I would do things a little differently.

First, standards-based grading! I started this in second semester last year, when we began studying series. It was a natural break to begin something new, but the content turned out to be perfect for the kind of self-reflection and improvement that I needed from students. (I’ve been implementing various versions of SBG every year I’ve been teaching; this might be my favorite version so far, but still far from perfect.)

Learning Objective 4.1A – Determine whether a series converges or diverges.

This learning objective encompasses A LOT of different tests for convergence, and what I found is that our textbook gave very bare answers to homework problems, so students thought they were doing great when they were actually missing big chunks of explanation that would communicate sound mathematical reasoning (and let’s be honest, would cost them points on the AP exam). By changing my grading strategy, I was able to better communicate my expectations and give feedback for improving.

After our first quiz on the topic, I gave students their highlighted quizzes back (no points marked, my typical M.O.) along with a Google doc for reflection. This included a rubric for them to score their own progress on this learning objective.

Rubric series

I showed an example of a 4 to the whole class, emphasizing three parts to communicating whether a series converges: conditions, evidence, conclusion. Many students were providing the evidence, but not first checking that the conditions for using the convergence test have been met or providing an appropriate conclusion; we looked at two examples of 3s that illustrated these omissions. Then students made corrections to each problem on their quiz and gave an overall score for their understanding with evidence to support their score. Ultimately they were very accurate and I only had a couple of conversations with students whose evaluation of their learning was different from mine.

The best part – every student earned a 4 on their next assessment of the same learning target. After a whirlwind first semester, I finally felt like I was doing something right!

I always give cumulative quizzes/exams, so this quiz also included derivatives from first semester, along with a rubric for self-evaluation similar to the one above. I got some very honest, “I thought I was a 4, but this evidence says that I’m currently at a 2” evaluations. They realized for themselves that they still had some work to do on the “easy” stuff! I really hope to start this from the very beginning of the year when/if I get to teach this course again… I think the potential for continuous learning here is huge.

The other thing that I would really want to do is address the sequence of content; I still don’t have a good feel for this yet. In my other courses, I try to space out related content so that students have time to practice content before trying to build on it. For example, in EM3, we teach unit circle and trig functions (all no calculator) in unit 2, practice it throughout homework in units 3 and 4, then unit 5 brings in inverse trig functions after they have built up fluency. I don’t know yet what this looks like in Calculus, but I would like to attempt it. Thoughts so far:

  • Do some work with series earlier in the course. Chapter 10 in our textbook is all about series and it is HEFTY. It could easily be two units with all of the convergence tests taught earlier and practiced before needing them to test endpoints for intervals of convergence later.
  • Teach derivatives and integrals simultaneously??? Maybe start with graphical and tabular interpretations of both, then move to analytic approaches? Can this be done??? I have more questions than answers.
  • I felt like I needed more time for teaching and practicing integration techniques, so find a way to do this earlier. My students sucked at u-substitution, honestly because I think I sucked at teaching it. Maybe by teaching chain rule and “backwards chain rule” together, students will become more fluent. Maybe by separating it from other integration techniques and letting them get good at one technique at a time, they can feel more equipped when I ask them to compare and contrast strategies. Maybe Coaching Questions would be a good routine to use practice using integration techniques.
  • Polar area – I feel like I can sneak this in earlier with other area integration problems. Still not sure.
  • The construct of teaching all AB content first, then moving to BC content, seems forced. Is there a reason for doing this, other than my textbook said so? Yes, the AB content is foundational for BC content, but can/should it be better integrated?

And there we have it folks – a lot of partially formed ideas about AP Calculus BC! I really do hope to teach this class again. Shoutout to my department chair – if you’re reading this, in case I haven’t said this enough, I would LOOOOOOOOVE to teach AP Calc again in the future. 😉 😉 😉

Posted in Uncategorized

Summer time = idea time

Blogging is hard. I really admire people who can carve out the time to do it regularly! I haven’t found a groove yet.

But one of my goals for the next year is to give myself more grace. There is so much that I want to do that I don’t always get done and I feel bad for it. Instead, I’m going to try to read more often, play with my kiddos, watch movies with my husband, lay in the hammock with a glass of wine, and try not to let all the things I “should” be or “could” be doing to do creep into my head.

Because there’s a lot of things! Summer is when I start to dream about the new school year: setting up new routines to make my classroom run more smoothly, inventing far too elaborate projects, trying yet another new way to implement homework that supports students and doesn’t make me crazy, etc. etc. Next year, I’ll be teaching 4 different courses, so I have even more ideas than usual swirling around in my head.

In my classes, I try to encourage students to have a “rough draft mindset.” Math thinking can be messy as we’re playing with a problem and trying different strategies. When we find our pathway through a problem, we edit and revise, seek feedback from peers, and rewrite our solutions to make our thinking clear and coherent to the reader. Now it’s my turn to employ this same process for planning for next year! I thought I would start writing about all of these ideas I have swirling around in my head, and turn some of these half-baked ideas into a whole series of blog posts. I’ll seek feedback from peers, and maybe… just maybe… they’ll eventually turn into real live lesson plans! (GASP!) Here are some of the ideas that I hope to expand on in a series of posts:

This year will be the third year of existence for Enhanced Math 3 (think Math 3 + Honors Precalculus all in one school year) and my third year teaching it. I’m really proud of all of the work I’ve put into this course and I think it’s a really great course! My goal for next year is to move to standards-based grading. I have convinced my colleague that this is an interesting idea, but he’s not ready to move forward with implementation quite yet. We’ve landed on a compromise that I like a lot! We’ll have students track their progress and use evidence to argue for the grade they think they should have.

I love teaching Math 2. I taught it two years ago when I first came back to teaching full time and I was disappointed to not teach it again last year. But I’ve used up all of my ideas for this course already! 🙂 I think this year will be more about refining and improving already existing structures, like standards-based quizzes, Ready Set Go homework, and spacing units for mastery over time.

AP scores just came out and I’m really proud of my AP Calculus BC students! I have so many ideas for what worked well last year and how I want to improve… I want to expand on the SBG structure I tried second semester, reorganize units, and rely less on the textbook and more on bigger conceptual ideas, like my first day of school lesson that paid dividends the entire year! Too bad I’m not teaching it again 😦 but I want to record all of my ideas so I don’t forget for when/if I have the opportunity to teach this course again!

I’ve already mentioned standards-based grading a lot. I have been implementing and revising various versions of SBG for 12+ years, since before I knew it was called standards-based grading. It’s a passion of mine and although I realize that it’s not the cure-all for problems in education, I have seen such tremendous transformation in math instruction at our school after implementing these principles. I could write 100 blog posts about our journey and how much I love what we do at my school.

The class I may be most excited and nervous about is our brand new Math Foundations course, which will be offered concurrently with Math 1. We have eliminated our Math 1AB course, which was originally designed to be a two-year Math 1 course (Math 1AB + Math 1CD); we eliminated Math 1CD a few years ago, so students who take Math 1AB then take Math 1 the year after. Starting this year, every ninth grader will enter high school taking Math 1 (or above), and those who have been identified as needing additional support will also take Math Foundations. This will be more structured than our Math Lab class (which I’m also teaching) which will be geared towards Math 2 and Math 3 students this year.

Of course, how could we start a new school year without talking about the first week of school? I plan to continue the good stuff I’ve done (thank you Sara van der Werf) and sprinkle in a few new ideas (thank you Howie Hua). But I also don’t want to overwhelm myself and my students with too much.

With that, my summer planning series is born! Even just thinking about writing about planning has helped me clarify my thinking. I look forward to writing my rough draft about my ideas, seeking feedback from peers and making revisions, and making the final product in my classroom next year the very best it can be.