Blogging my Guerrilla PLC

My district has mandatory PLCs but I hate being told what I have to do to grow as a professional. Anyhow, I am rejecting the official PLC and going guerrilla (which is my understanding about how PLCs work best anyway).

So, I'm teaching Algebra 2 for the first time in a long time. (Not to get too far off topic here, but I hate it when teachers say, "I haven't taught that in 10 years. I don't know any of it." Really?! You don't remember the big ideas of Algebra 2? I haven't taught it in 10 years and I'm pretty sure that it hasn't changed that much. I get that you might forget a formula here and there, but the entire curriculum?You obviously just memorized it and didn't really know it in the first place.) I remember that Quadratics always give kids a hard time. And, I know that other teachers feel the same way. Fortunately, Twitter to the rescue! James Tanton (who I've reference elsewhere) has a free, online Quadratics course.  I asked a few colleagues to join me in watching some videos and trying to create a good, cohesive, conceptually based unit on Quadratics. 

So, we made some goals: (1) We need to create a good unit on Quadratics. (2) Students should understand something about the history of Quadratics, it helps to ground the unit. (3) We're going to watch/do everything ourselves but will probably adapt for our students.

This might not be the best goals so far, but I think they'll get adjusted through the process. 

Then we got down to work. Watched a few (pretty awesome) videos and learned some (pretty awesome) math. Some of it I had seen before, some of it was new to me. I'm excited about the possibility of making something fun and interesting for class. We didn't make it through all that we intended because the practice problems were pretty challenging, both computationally and conceptually. Then we gave ourselves homework. We have to watch three more videos each and do the practice. When we get together next week, we can go over stuff together, share what we've learned and then go on to the next section. 

I'm pretty pumped. I think this is going to be awesome. And I think to myself, isn't this what teachers in other countries do all the time? Why did I have to carve out personal time to learn something for my job? Whatever... my new motto: when it comes to professional development, go guerrilla!

Low Expectations for Algebra 2

First off, I really don't believe in natural ability. When I hear teachers saying things like, "These kids just aren't math people", I want to scream. I understand that there are gifted people out there in the world, but hard-work trumps talent any day. Many kids are mistakenly led to believe that they aren't "math people" by elementary teachers who think that there are "math people". Instead of encouraging productive struggle, we let kids with learning challenges off the hook and make them think that they're stupid. By the time they get to high school, teachers are fighting a battle whose outcome had mostly been decided by a teacher who got to the kind long before you did. It's an unfortunate state of affairs. 

That said, my Algebra 2 class this year is chock-full of students who consider themselves weak at math ... and, truthfully, they are. I haven't taught Algebra 2 in a few years but I've been in conversation about it with other teachers in my school. In my grand plan for what my Algebra 2 course would look like, I figured that I could run through a quick review of linear functions and systems of linear equations and then plunge headfirst into quadratics before Halloween. Well, Halloween is next week and I just gave a unit assessment on Linear Functions. Now we are going to delve into systems of equations ... but first I thought I'd do something investigative.

I feel like I have low expectations for the class. A large percentage of the class are classified Learning Support, which really doesn't bother me very much. And I also don't buy into the idea that I have to go slower for them. Well, maybe slower, but not less rigorous. All students should have access to high quality mathematics. (Do I sound like an ad for the CCSS-M?) I found a good activity in a recent Mathematics Teacher which compared miles per gallon and gallons per hundred miles. I prepared for a large group instruction/discussion and expected the worst. I was sure that I was going to get a barrage of, "this is stupid" or "this is too hard" or "this doesn't make any sense" or "are you going to the football game on friday". Pretty much anything except a high quality discussion about these ratios/measurements. 

What happened did surprise me. The class was quiet and paying attention. Admittedly, there were only about 4 students actively answering questions, but the remaining ones were either working independently or following along and making sense as we went. There was a lot of head nodding and button pushing and I'm pretty sure that by the end they had a good idea of where we had been and picked up a few skills along the way. We still had to end the discussion 5 minutes before the end of the class because they can't concentrate for 45 straight minutes... but really, who can in 2013? 

The rest of the discussion, and the real meat of the activity, is tomorrow. If it goes half as well as today, I'll be pretty ecstatic. 

Why do I need to learn this stuff?

Just so that everyone knows that I'm not a total complainer:

A few weeks ago, one of my 9th grade students asked me the question I hate to answer: "why do I need to learn this stuff?" First, I gave the terrible reason that he has to know it to pass the state-mandated test at the end of the year. But, because I believe that there's a lot of superfluous content in high school math, I also told him that he might never need to know it ever. I went on to say that because we cannot predict the future, it is impossible for me or for him or for anyone else to know whether or not he will ever need to know it. And, often, when adults are asked, they say that they wish that had paid more attention in school especially in math class. By now I was on a roll so I stepped up on my soapbox and continued. Mathematics is not just about getting right answers. Mathematics is a way to experience the world and a way to think. We, as humans, do two things that are different from all the other animals on the planet: we explore the human condition through history, literature, art, music, etc and we attempt to understand the natural world through chemistry, biology, physics, mathematics… Mathematics helps us think logically and creatively. It gives us a way to explain and explore the patterns present in the universe. Without mathematics, our 21st century way of life does not exist. Without mathematics, beauty does not exist. 

At the end of my ranting and raving, another young man raised his hand and said, "Most of the time, when you ask a teacher that question you get a lame answer. But that one was pretty awesome." I hope it sticks with a few of them.

Formative Assessment

Each year, my district decides on a focus for the professional development. This year it's formative assessment. I like the idea of formative assessment. I like doing un-graded exit problems, or asking a question and then choosing a student's name from a shuffled deck of index cards, or Poll Everywhere or Bell-ringer/Warm-up questions or even something as mundane as a practice quiz. I don't use it nearly  as much as I should, but I like it.

Here's the thing that confuses me: Isn't formative assessment just part of good teaching? If you're just standing up there talking at kids and only taking the time to figure out what they know/don't know, what misconceptions there are, what the pre-existing knowledge base is, when you've finished talking, then you're not really doing your job. The advantage of having a real, living, breathing classroom teacher is the ability to collect data and make near-instantaneous changes to improve the learning experience. This is something that a virtual presence (read: online video tutorials) can't do right now.

But, I'm over the fact that someone who probably doesn't know as much as I do is going to tell me how to start incorporating formative assessment. The issue I have is that my district is also really big into conformity right now. What I mean is, everyone has to use the same formative assessment. What is the purpose for this? No one can give me an answer beyond, "Everyone should do the same thing" or "Collaboration is good". If we really want everyone to do the same thing, then we should plop our students in front of on-line videos and hope for the best. I agree the collaboration is good, but in my experience, a group of more than 4 math teachers who have been assigned to work together have too many different philosophies of education and mathematics to create a usable product. 

Moreover, I believe that formative assessment is supposed to give the teacher information about how things are going in your classroom. It should be short and only on 1 topic (2 at most). You need to read the answers yourself to get any information. And the kids should get them back the next day (at the latest) so that they can use your feedback to progress.


These are the things I overheard today in the teacher's room:

"I had the students grade each others' papers and then I wrote down the score."
WHAT?! How are you (the teacher) supposed to get any information if you didn't read and evaluate the answers yourself? You've missed the whole point.

"I don't understand why it can't be a minor grade for the student."
WHAT?! The whole purpose of formative assessment is for you (the teacher) and them (the students) to get some information about who knows what and how well. If it's a grade, then you are potentially punishing students for not being proficient when you should help them learn it first. And don't get me started on how ridiculous grades are anyway. 


I'm going to keep doing formative assessment because it's helpful and it's important and it's revealing. But I won't give the crappy assessments designed in committee because they're crappy and because they're not going to tell me anything worth knowing. 

The germ of an idea...

Full disclosure: I read a lot. Sometimes novels, sometimes magazines, sometimes YA, sometimes SciFi,  sometimes philosophy, sometimes books about teaching/education, sometimes books about math, sometimes blogs, sometimes Twitter... pretty much anything (except historical fiction... gotta draw the line somewhere, right?) The point is that when I wonder about something, my default position is to do some reading and then reflect. This post is mostly reflecting about teaching math as it pertains to a specific pedagogy.

I've been assessing my own teaching lately and been pretty down about it. I lament the fact that I can't seem to get my 9th grade students to engage with Algebra 1 beyond a "how-to" level. I know it's a process and I know that it'll get better if I plug away at it... but it seems so far away.

There has been some success: I just have to remind myself about a student from last year who told me that the only reason that she passed the state test was because when she got to a question that she didn't know how to do, she thought to herself, "What would Mr Mango do?". It was very nice of her to say, but of course, I don't think I deserve that much credit.

I digress: I've decided that I need to re-root myself. I'm going to re-read Bob & Ellen Kaplan's book, "Out of the Labyrinth".  (Christopher Rice, I love you, but you're going to have wait a few weeks. I'll get to you.) I've been to a few of their training sessions and I've totally drunk the Kool-Aid on Math Circle techniques. But. What Am I Doing Wrong?

I think it's that I'm not starting the lesson (planning or instruction) with a question. Bob and Ellen always start their Math Circle sessions with a question; a place from which inquiry can venture forth. I've known for some time, but maybe haven't verbalized the fact that all of the crap stuff that we teach kids in HS math is the answer to some question. The teacher's job should be to ask the questions and help kids see where the answers come from... not to tell them the answers and imply that the questions are unimportant. No wonder math is boring for most kids/people. We don't ask enough questions.

Why do kids always ask, "Why do I need this?" or "Who made this up?" or "What is this good for?"? Well, not entirely absent from a lot of teaching is the inquiry that precedes the mathematical technique. I'm not asserting that no one will ever ask again "When am I ever going to use this?" but starting with a question might help alleviate some of that. Resolution: I will incite as many lessons as possible with a question.

So, I'm embarking on "Out of the Labyrinth" again... for the 4th or 5th time. New insights this time? Pretty sure that a given.

Motivating Taylor Series

I'm teaching AP Calculus BC for the first time this year. I was sort of excited about it. I taught AB calculus last year and in the school where I teach, BC students have already taken AB. Which means that I get a bunch of (really awesome) kids for second year. Then there was the realization that there actually isn't that much new content that is BC only and the stuff that is new mostly fits into the story you already unfolded last year. At least, that's how I feel right now.

Of course, then there are Taylor Series. I only vaguely remembered these from my college courses, and somehow I managed to avoid them in grad school. My recollection is that I had no idea what they were or why they were used. And then there was the fear that I'm going to be forced to, the horror!, tell my students how to do something instead of getting them to explore the idea. So, I took an AP summer institute and it assuaged many of my fears. The instructor was good and did develop the idea of Taylor series and motivated their use. I was pretty excited about it. 

Then I remembered that I pretty extensively used James Tanton's Thinking Mathematics (Vol 6) in developing the AB course that I really liked to teach. Fortunately for me he also has a Volume 7 for BC calculus! He has an excellent chapter (actually, he has lots of excellent chapters, but for the purposes of this lesson, there's only one) on motivating series. It goes something like this:

A lot of functions are difficult to evaluate, but polynomials are not one of them. Maybe, I wonder if, we could develop a polynomial that could be used to approximate the values of a not-nice function. He leads the reader through the development of the polynomial, which is really just a series).  But what makes Tanton's books awesome is that you feel like he's trying to engage you in a conversation. So my job is taking that conversation apart and making it an activity for students to explore. Right now I'm thinking that I have small groups of 3 or 4. Each group gets a different function (sin, cos, e, ...) and get them thinking: 
What would this mystery approximating polynomial look like if the function and the polynomial only agreed at 1 point? (Who knows? Couldn't it just be a horizontal line?) 
What about the slopes? Should the slopes agree at that one point? (That makes sense... isn't that a derivative?) 
How good is a tangent line at approximating the values of the function? (Good in a small range around the tangent point; pretty bad beyond that.) 
What if you tried to make the function and the polynomial have the same concavity too? (Really? Second derivative? I guess I can try...)

Look at the graph of the function and the approximating polynomial. How close are they? Does the polynomial do a good job of estimating the points? By how much? When does the polynomial start to fail? What do you think you could do to get a better polynomial?

Then jigsaw the groups and get some similarities and differences going.  

That's my idea right now. It'll go through many mental iterations before it goes through a few print alterations. But I'm looking forward to it.

"Well, you're a numbers guy."

Actually, no, I'm not. I'm not very good at arithmetic and the very idea of statistics (beyond high school level) makes me queasy. I find the theory of numbers fascinating, but that's not the same thing as being a "numbers guy".

The assumption made by many of my students and most of the people I meet and a lot of the folks that I interact with daily is that I love everything to do with numbers because I teach math. The numbers part of math is what I find the most boring and mundane. It's the ideas that count. The ideas, the concepts are amazing. The numbers? Hate them.

And maybe it's not really so much the numbers as it is the assumption that I both love and excel at arithmetic. I don't. When the check comes when I'm out to dinner with friends, someone always glances my way like I'm supposed to jump at the chance to perform some feat of computation. I routinely joke that I don't do arithmetic on the weekend. Mostly I'm trying to save myself the embarrassment of getting the answer wrong. Then I'll have to endure the taunts of, "You're a math teacher and you can't divide 102 by 7?" You know what? I can't. And I'm OK with that. Somehow, I've managed to graduate from college, get a series of teaching jobs at both secondary and post-secondary levels, acquire a master's degree (in math) and get so close to the completion of a doctorate (in education) that I can almost taste it and I can't do a lot of mental math. But I can help kids learn the ideas that really make up math and let them know that it's OK to suck at arithmetic if you understand the underlying concepts. We have machines that are much better at computations than humans will ever be. But those machines are pretty bad at identifying similarities and creating analogies. That's where a person comes in.

Why I love the idea of SBG...

Let's face it: grading is hard and grades are mostly arbitrary. I love when I go to an IEP meeting or I'm in a PD session and I hear a teacher say something like, "They got a 72% on that quiz". A series of questions always come to mind: What made it 72%? Did the student acquire 72% of the possible points? Did the student show that he/she understood 72% of the content on the assessment? Did the student answer correctly 72% of each question? Can you (the teacher) really show that the student knows 72% of the test? How do you know that it's not 70% or 75%? 

I didn't always think this way. When I was a new teacher, I graded the way I thought I should... which was probably the way that I was graded as a student. Each question was assigned a certain number of points and I subtracted points based on mistakes. Eventually I started to think about grading as adding points for correct work instead of subtracting points for incorrect work, but I was still pretty discontented. Suppose that I had two equations on a test that were worth different points, say 3 points and 5 points. I usually counted an arithmetic mistake as 1 point. Then, the same arithmetic mistake in each problem was worth a different amount for each question... ~33% versus 20%. Was an arithmetic mistake really worth 33% of the first problem? 

What really brought my thinking to a head was when I was helping a student that I had taught in 9th and 11th grades with his senior year math class. I knew from my experience that he struggled with entering computations in his calculator correctly, but you could tell that he had a pretty good grasp of concepts even if his arithmetic was bad. On one test there were about five questions that looked something like, "if f(x) = 2x + 1, find f(3)". He set up every problem correctly, then plugged them into his calculator wrong. The teacher had assigned each question two points and took off one point for his arithmetic mistakes. I thought to myself, "This is crazy! Does this kid really only know 50% of each problem?" Obviously not. 

After some serious searching, I stumbled upon Standards-Based Grading... probably from Shawn Cornally. I thought this would solve all of my problems! A rubric for each question! Topics scored instead of arbitrary textbook sections/chapters! This would be perfect! I could really get a picture of what kids know. I would be able to communicate qualitatively with parents and students about strong and weak areas, not about good and bad grades. SBG & Rubrics would be my saving grace! 

Whenever I explain what SBG is to a teacher/parent/student they mostly think it's a good idea. There are a few exceptions. I've been fairly successful at implementing an SBG system in my AB and BC Calculus courses. The other courses I teach? Not so much. 

Coming next time: Why I had to (mostly) abandon SBG...

First Post

I'm writing this blog, not because I am simply discontented with the state of public education, the lack of student interest and my own recent decent into what I would consider sub-par teaching. I long for something better. I know that I will sometimes come off as ranting and complaining, but that's not the focus. The goal is to help me make sense of what I yearn for and how I might be able to make some substantive changes.