Three lessons

One of the most important lessons I learned about teaching I learned as a graduate student working in the Math Lab at my university as a tutor.  We had stations where the tutors would sit (and, quite frankly, work on our own homework) until someone approached us with a question.  At this point, we would gladly and enthusiastically put our homework up and help answer whatever question the undergrad student had.  We thought we were so approachable and were awesome tutors.  Or at least I did.

Until we got a scolding from the head of the math department.

Apparently, I was not as awesome as I thought I was.

Our boss (who is one of the kindest human beings on earth, I might add) gently told us that maybe we weren't quite as approachable as we thought we were.  "Math is really intimidating for most of the people taking these classes.  It takes a lot of courage to get up, in front of everyone, and come over to your station to ask a question."

I'm paraphrasing as it's been several years, but that was the gist.

He encouraged us to go to them.  I remember feeling so humbled.  Of course, he was completely correct.  As soon as I started making my "rounds," the amount of questions I got each day skyrocketed.  Furthermore, I started building rapport with several of the students who came consistently.

This experience greatly shaped the way I now teach high school math.  I'm very against sitting at my desk and letting students come to me.  Because that's what I did as a TA in grad school and it clearly does not work.  You know who comes to ask questions?  The kids who are going to figure it out with or without me.  The resourceful ones.  The ones that need me the least, to be honest.

When kids are in the room, I believe they need to be my primary focus--not lesson planning or grading or writing a quiz.  When kids are with me, they must take precedence:  they are reason I'm there, after all.  This philosophy means I've created methods to grade homework as they go (these methods vary with each of my preps) because prioritizing means something's gotta give.  For me, that's homework grading.  I'd rather spend my time with the kids than grading their homework meticulously every day.  The rest--lesson planning, grading quizzes/tests/projects, writing quizzes/tests, writing rec letters, etc.--that all happens when kids are not in the room:  during my plan, after school, or during the weekends.  That's how I've decided to prioritize and manage my time.  Everyone's different, but my main point is:  our kids need us when they're in our rooms.  So whatever you have to cut out to make time to be with your kiddos, I think it's worth it.

This brings me to the next important lesson I've learned as a teacher.

While I'm pretty good about making my rounds and staying away from my desk (on most days...I'm not going to pretend I'm never at my desk during class time), one of the things I've practiced more recently is being able to pull questions out of kids.  During my rounds, I would often ask questions like, "How's it going?" or "Can I help with anything?"

I thought those were perfectly fine questions.

I assure you, they are not.

I've replaced those phrases with "What questions do you have for me?" or "What may I help with?" or "Tell me about your thought process here."

Goodness.  What a difference.  I cannot even begin to describe how many more responses I get when I invite questions in this manner.  It calms the kids when I approach them with an air of "I expect you to have questions for me, and I want to help you reach a deeper level of understanding."

If you're not convinced that these questions are all that different, take this anecdote as an example.

I approached a kid a couple months ago and asked him, "How's it going--can I help with anything?"

"I'm good!"  he responded with a smile.

I was tempted to leave and move on to the next student, but I knew I owed it to him to pry just a little deeper.
"What can I help with?"

"Actually, could we talk about Number 7...?"

As a teacher, the two questions I asked should mean the same thing.  But to students, they clearly elicit different responses. 

The last important method I use on a daily basis is also very simple, but I believe it's really powerful.  When I help students and I know it's going to take a while, I get on my knees right next to them (or, if the seat next to them is open, I might opt for that). I do this even if I'm wearing a skirt.  Even when I'm eight months pregnant.  It's a way for me to physically say, "I'm here to serve you.  I'm not going anywhere."  I believe this small and simple gesture has broken down so many walls.  It's impossible not to be touched by humility.  

Those are my three lessons.  I typically try to stay away from giving advice (I think most people just need us to listen more than talk).  But, these are lessons that I have to intentionally practice every single day.  It's advice for me as much as it is for anyone else. I hope, though, that it helps others, too. Or at least helps others form their own welcoming classroom culture.  

Derivatives of Inverse Functions

This is my fourth year teaching calculus on some level.  Every year (until this one!) my students have really struggled with finding the derivative of inverse functions at a point, especially in the manner these questions are often phrased on AP Exams.

To me, they're some of the most straight-forward multiple choice questions the students encounter on the exam; yet, year after year they miss this question (at least on their unit tests and mock exam).

So, clearly, not as straight-forward as I thought...

This year I formalized a strategy for them in three steps.  Not all three of these steps are necessary every time; but, if my students took the time to follow all three steps, they got these questions correct.

Here are the steps:

If f and g are differentiable functions and g is the inverse of f, then to find g'(a):

1. List all points given on f as ordered pairs.
2. List the points you now know are on g (switch x and y).
3. Follow this formula: g'(a)=1/(f'(g(a)).

Let me show you with a couple examples.  Here's a question I pulled from this website.

Following the steps, we would work this question as follows:

How about one that describes f as an algebraic or numeric function, such as this FRQ from 2007:

Students could certainly start with Step 1 again and work their way down, but I encourage them--once they get comfortable--to feel free to start with Step 2 and fill in the blanks as they see fit.  Here's how I would suggest they work this problem:

That's it!

The Peterson Diagram

Here's a short (very low-tech) explanation of a diagram I created a couple years ago to help my students answer questions about how f, f', and f'' are related.  My kids use this especially when--for example--they're given a graph of f' and they have to answer various questions about f, which seems to be a favorite of AP Calculus Exam writers.