Reflections From #precalcchat: Pre-Calculus Sequencing

I love Twitter chats with other teachers. It's a great way to make connections. It's a great way to get insight, ideas, and resources. It's also a fantastic opportunity to reflect on your own practice and to improve what you're doing in the classroom.

The Global Math Department hosts several weekly Twitter chats for math teachers on a variety of topics. Since I teach Pre-Calculus, I dropped in on the first #precalcchat of the school year last week; thanks to Mimi (I Hope This Old Train Breaks Down...) and Taoufik Nadji for hosting. Couldn't spend much time, but the topic of conversation captured my interest:

I loved that thought. It made me stop and think about how I sequence my Pre-Calculus course and why.

I start with Graphing and Functions first. To me, it's important for students to understand the basics of interpreting graphs of functions and becoming fluent with moving between different representations of a function (graphs, tables, equations). I find this to be a particularly vital theme that I want to drive home with my students, especially those who will be going on to AP Calculus or Calculus I/II in college.

Next, I follow a pretty standard sequence of Quadratics/Polynomials, Rational Functions, Exponential Functions, Logarithmic Functions, Trigonometric Functions, and Analytical Trigonometry. Again, I focus on these topics in particular to prepare my students for success in an AP Calculus course. Other topics such as Analytical Geometry, Series & Sequences, Polar Systems of Coordinates, Conics, etc. come afterward as time allows.

The other chatters all had brilliant things to say, so naturally I felt like I'm probably doing everything wrong (or maybe just some things wrong, and other things not-as-wrong).

When discussing how Pre-Calculus can seem like a re-teaching of Algebra II to students, Tina C (Drawing On Math) mentioned that her school starts with Trigonometry for that exact reason.

This was an interesting idea to quite a few of us: do Trig first semester, slowly build up conceptual understanding of the unit circle, graphing, transformations, identities, etc. Then, move into the other different functions second semester.

The more I think about doing Trig first, the more appealing it seems to me. I've always found that I never seem to have enough time to really properly teach Trig and I need to either rush a few things or cut some other stuff out. I think I probably always had the notion that Trig is "more difficult," and somehow it made sense to put the "harder stuff" at the end of the year. (That's excellent reasoning, isn't it?)

But really though, Trig is a bit of a stand-alone topic. It could go anywhere in the course sequence. There are certainly some underlying concepts that can be applied to other functions: graphing, transformations, moving fluently between representations, and so on. I usually think of these concepts as having to be taught and mastered before doing Trig, as if Trig is the "CHALLENGE MODE" of working with functions in Pre-Calculus.

Who's to say we can't use Trig to teach these concepts instead? Maybe my students would have greater success with Trig if I did it at the beginning of the year, built the concepts slowly with appropriate scaffolding, while still equipping students to be successful in working with other functions. I may have to try it out one of these years. (I already have this year mapped out -- maybe next year?)

Anyway, some great food for thought.

I'm looking forward to more of these chats this school year, and hopefully I'll find time to continue blogging & reflecting on what I take away from them.

Week Zero: Realizing I Might Actually Know Stuff

It's Week Zero. School Year Eve. The last few days of summer before I get to go back into the classroom and spend the next nine months convincing teenagers that math is freaking awesome.

I'm a teacher mentor this year, which still seems crazy to me because I'm only four years into this profession myself. On Monday, I went to an all-day mentor training session to learn about my role and responsibility as a mentor. A lot of the information was about what I had expected: the mentor wears many different hats, has to build a relationship of trust with the mentee, can learn just as much about teaching from the mentee as the mentee does from them, etc. and so on. We talked about how to have positive conversations with our mentees, how to listen and to provide feedback, and best mentoring practices in general.

We also got toys and candy, which was super cool:

One thing that struck me from the mentor training was what distinguishes a good mentor from a not-so-good mentor: the desire to keep getting better as a teacher. Good mentors know that they still have things to learn about teaching, and no matter what the difference in experience is, they can learn a lot from their mentees. (I'm pretty convinced that I'm going to learn more from my mentee than my mentee is going to learn from me.)

I was reminded of this the next day (Tuesday) when I attended the first-day morning session of new teacher orientation. I sat with my mentee throughout the morning as we introduced ourselves and learned various things about the teacher-mentor program. We had time to talk about the upcoming school year and I was able to answer some questions about curriculum and how we do things at our school.

The experience made me think back to my first Week Zero in our district, when I went through new teacher orientation. I remember feeling excited and nervous about my first year of teaching. I also remember thinking that I was probably going to make a lot of mistakes, I was going to have to learn from them, and there was so so much about teaching that I didn't know yet.

I had the same excited, nervous feeling this week. I still feel like there is so so much about teaching that I don't know. But, in the act of answering my mentee's questions, I was struck by another thought: I actually, maybe, perhaps, do know stuff about teaching now. I had never really thought about it until someone else was asking me. When I was answering my mentee's questions, I really had a lot to say. I had a place of experience to speak from. Holy crap, I have experience. And it might even be useful to someone else.

That might be my important realization from this week: There are many things about teaching I still don't fully know. But I'm also starting to understand how much I do know about teaching. Maybe I'll actually be a decent mentor.

Anyway, back to work! Students come back next week!

Never Be (Fully) Satisfied

The past few days, I've been reflecting on how much time I spent this summer working on writing and tweaking curriculum for the new school year. It's not exactly a new activity for me -- I pretty much write and tweak curriculum every summer -- but I think I probably got more done this summer than I've ever managed to.

I actually fleshed out two different curriculum maps with topics & aligned standards (first attempt at aligning Common Core, so probably lots of mistakes). I'd never made curriculum maps with a great level of detail before, and I'm pretty sure I'm going to be very thankful I did so this summer.

I also spent a lot of time this summer working on incorporating more Problem-Based Learning (PrBL) tasks & lessons into the curriculum (one such idea I had is detailed here; feedback is more than welcome!). I teach at a New Tech Network school, so a rigorous PrBL curriculum is my goal. I've spent hours and hours looking for ideas, researching, thinking, scribbling in my notebook (particularly for those middle-of-the-night ideas), typing pages of details, and probably making my wife very annoyed that I was spending so much time working. I hope the result is that my students do some really awesome, really meaningful learning this year.

Another goal of mine is to learn more about Common Core (I admittedly am still a novice), so when my principal e-mailed the staff earlier this summer about attending a Common Core workshop in September, I was all like "MEMEMEMEMEME!!" So, I'm excited to go, learn some more about Common Core, and hopefully take away valuable knowledge that I can incorporate into my professional practice.

And the idea of improving my professional practice is something I've been thinking about over the past few days.

I've found myself thinking a lot about all the things teachers do to try and improve their teaching. I see many teachers who I follow on Twitter talk about all the conferences they attend and share what they've learned. I have several friends who are enrolled in masters programs, learning more about educational technology, developing curriculum, or otherwise broadening their skill sets as educators. I've thought about the things I've done each summer since I started teaching: working on curriculum, participating in the professional community, working on my own masters, and constantly thinking (and often worrying) about how I can be a better teacher.

And as I thought about all of this, I realized something: I'm not sure I ever want to be satisfied with the kind of teacher that I am.

I'm sure not satisfied with my teaching right now. Frankly, I'm not that great at it. (Sure, I'm funny, handsome, irresistibly charming, and very humble; but from a pedagogical standpoint, those traits can only carry me so far.)

But I don't think I want to ever be fully satisfied with my teaching, not even after I've been teaching for thirty (forty? fifty?) years. Sure, I want to feel happy about my teaching, which I think is a different thing. But not satisfied.

I think it's probably easier to feel this way now, since I'm only going into my fifth year. I know that I have a lot more to learn about teaching. Any fool can see that. There are roughly eleventy billion areas where I can to improve my teaching. I have rather lofty goals for myself this year. I might not meet them all this year, but that just means I'll regroup next summer and try again the following year. And the following year. And the following year. And so on.

But when I've been teaching for a few decades, I don't know how easily I'll still see all of that. I don't know if I'll still be this enthusiastic about improving my craft or if I'll be like, "meh, I've been teaching for thirty (forty? fifty?) years, I'm awesome enough." I don't like that idea. I really hope instead that I'll always want to be a better teacher than I was the year before. Even if it's just a teensy bit better. My students deserve that much, I think.

I talked about this with my wife the other evening. She understood where I was coming from, and noted that this is true about many professions. I mentioned that I was (and am) nervous about meeting my new students on the first day. She said one of her past supervisors once told her that's normal; "that means you care." And my feeling nervous doesn't really stem from being scared about meeting a new group of people, but more from really really wanting to be a better teacher this year than I was last year. I don't want to let these kids down.

Summer is great. It's a time when teachers can work on improving themselves and do what they can to make the next school year better than the last one. I spent a lot of time this summer working on that. I always want to be doing that. I want to be happy with who I am as a teacher. But I also think that I want to never be satisfied. Maybe mostly satisfied. But not fully satisfied.