I’ve taken away a lot of valuable ideas from the Klingon summer program. I wanted to distill them into a really strong set of reflections, but I’ve found I haven’t been able do that because I am still trying to play and tease them and turn them into some sort of coherent philosophy which I can envision being enacted in my classrooms. So as opposed to most of the blog posts I tend to write, I’m going to go a bit theoretical, and ask for your help to come up with concrete ways to show these things happen.

The first thing I’ve been struggling with, and I now see as one of my biggest weaknesses as a teacher, is the lack of attention I’ve given to bridging the gap between “the taught curriculum” and “the learned curriculum.” What’s strange is that I thought I had been giving that a lot of attention — and was one of my strengths. Here’s the idea:

We all have a plan of what we want to cover in a year, in a class, or in a small bit of a class. That’s our “intended curriculum.” Then we go about and teach stuff. That’s our “taught curriculum.” Then there’s actually what our kids learn. That’s our “learned curriculum.” It has been kinda obvious to me since my first year teaching that there is always going to be a gap between the taught curriculum and the learned curriculum. These commonplace catchphrases are things I’ve heard over and over in the blogosphere…

Just because you teach it, it doesn’t mean they learn it.

Teaching is not the same as student learning.

So the goal of teaching is to minimize the gap between the taught curriculum and the learned curriculum. And I thought I was doing that. I see myself as being cognizant of the mistakes that students tend to have in thinking, I work on developing their understanding rather than having them rely on algorithmic/procedural learning, I teach my classes at the pace of my students — and alter how far we get with the material based on how much time I see them needing.

However, I have come to realize that I have been missing a gigantorific thing, because I’ve had some sort of blinders on. Especially egregious because this has been the central idea and nexus of the math teacher blogosphere for over a year now. I’m embarrassed to say: I don’t have any idea if I’m being successful at reducing the gap between the taught curriculum and the learned curriculum.

In other words, if asked how I assess student understanding in the classroom before tests, I wouldn’t have a good answer. Yes, I do that whole I look at their faces thing (which tells a lot) (but some of them are good at playing the game with their faces too) and I do the whole check yo’self before you wreck yo’self thing (questions which have students solve a problem after we have gone over a concept and done one together as a class) and I do the whole let me ask you a question thing and I very occasionally do the whole “put your head down and raise the number of fingers” where 1 finger is ‘totes got it’ and 5 fingers are ‘what just happened?’ …

But when it comes down to it, if my department head came to my class three-fourths of the way through, and froze the kids (okay… okay… no… she can’t do that… only I wield that power with my magic unicorn wand!) and asked me “what is the level of understanding of each kid in the class for the material you covered yesterday?” and “what is the level of understanding of each kid in the class for the material you covered today?” I wouldn’t be able to answer her with much confidence.

I don’t have strong systems to formatively assess my kids understanding.

Yes, there you have it. I said it.

Now I need to find ways to do this. I am currently thinking of giving short quizzes at the start of every class and exit slips at the end of every class. I would like to pilot out clickers at my school. And I would like to come up with more intentional questioning which can get to the heart of whether a student understands a topic or not (meaning: I have these questions prepared before class and I can whip out when I’m ready to check their understanding).

But more than anything, I’d love to hear ways you come up with ways to formatively assess kids understanding during a class or even outside of classtime. Big, small, thoughts on how to form questions that get deep to the heart of the subject matter, ways to figure out what every kid or what most kids know?

For the past week and some, I’ve been at the Klingenstein Summer Institute for Early Career Teachers. We’ve been kept so busy that I haven’t had a lot of time to post, so I am losing insights left and right. Sigh! But I am making it a priority to talk about the reorientation I’m having with regard to math curriculum before I head off to PCMI.

One thing we’re doing today is having others watch videos of us teaching. And it’s terrifying. It was terrifying when I was being videotaped. I don’t know what will happen, how we will structure the conversations (they give us free reign to do what we want with regard to the conversations),

Something that I’ve heard happens with video analysis is that the teacher, before doing it, has one conception of who they are and what they are like in the classroom. They have this impression based on what’s going on in their head. When they watch the video, they suddenly see a very different person, someone they might be unacquainted with. It breaks you down.

That’s what happened when I watched my video last night.

But I’ve held, for about two years now, that I felt that video analysis of teachers is actually going to be the most powerful and next form of serious professional development… So I’m trying to overcome my fear, and so I asked a couple of our mathemablogotwittersphere cabal to watch my video and send me feedback. It’s terrifying. I feel like my blog gives one big false impression of who I am as a teacher (read this), and so it’s scary to think that people who I respect a lot will suddenly think: WTF this guy sucks.

But then I decided: heck if I’m not okay with that, because isn’t that the whole point of me blogging and twittering. I reoriented my thinking to say I own this class (I felt when I finished it was one of my better classes) and I’m using this to suck a little less. Going with that mindset (Dweck’s “growth mindset” anyone?) has made me feel like I can do this.

Anyone who has any experience with videotaping their classroom and self-analyzing their tape, or analyzing other’s tapes, or online resources which give a “method” to analysis, or any notes about their experiences, I’d love to hear them in the comments.

UPDATE: I just want to clarify, I am taking things slowly so I don’t think I’m ready to share it with more people, yet. i just want to slowly edge myself into this.

Just 5 minutes ago, I was taking a refreshing cold shower — because it’s too dang hot! And as best ideas are wont to come when paper and pencil are not around, I stumbled upon, in the rambling brambles of my nonlinear thought process, exactly what my summer project is going to be.

A little background first. The museum of math has had a series of math lectures (Math Encounters) this year, and so far I’ve been to all of them. They are delivered by people with grand speaking skills and on topics which are fascinating and excite the imagination. You can watch the first one (that they’ve put online) here:

I have decided I am going to try to come up with 3 lectures that I’m going to give in the first semester to students at my school. I’m thinking these are going to be after school things for anyone interested, and honestly, I will probably only get a couple teachers and a couple students to come. But it will be a fun lark for me this summer.

To be clear, I’m not talking about a workshop or problem solving sessions or anything super interactive. I’m actually thinking straight up lecture (with maybe some audience participation).

I’m excited enough about the idea that I think I will probably follow through on it. And so I thought I’d share the idea here, in case I can get others interested in doing the same. Anyone out there interested in doing it? I don’t think it would make sense for us to work together on the actual lectures, but I do think bandying about ideas for possible fun and high school accessible lecture topics could be superfun.

Just off the top of my head right now I have a few ideas: continued fractions, Farey sequences, the violent and sordid history of mathematics [I’d have to do some fun research on this one], topology, etc. Oooh! Coming up with good lecture titles will be EXTRA FUN!

My secret hope is that this is something that widens the scope of what kids think “math” is. I had that happen to me when I went to math camp in high school, where I was treated to so many amazing lectures on so many weird and fun topics that I saw the huge scope of math and saw the beauty even more piercingly than I had when exploring it on my own.