My personal biggest challenge in this regard is statistics. Stats demands precision, and I’m just not there with it yet. It’s a work in progress. If I am every talking about you with stats and I say something ridiculous, please feel free to correct me. I will appreciate it.

Beyond stats, function notation is one of those things that I know a lot of people are very casual about. I see teachers and students (and I know I’ve done it when talking) swapping between f and f(x) as though they are the same thing. To help with clarity, and the seed that made this post happen, here is a nice discussion about function notation that I am blogging about so I have it easily in reach and I thought some of you would also find interesting.

Oh, and there’s that Podcast thing that is now up. There will be a new one up this Thursday, 3/21 that’s a shorter format focused on a prompt, what to do about GReader, and other news of the online math world.

To listen to Episode 0: An Introduction and learn more about the podcast, click here. Thanks, and please share!

You can go and read the whole thing here. If you skip to page 20 you can read their “What We Know” conclusions from the 3 year study. I like a lot of what the report has to say about how to use classroom observation, rigorously training observers, taking a balanced approach, etc. I have a hard time with a standardized test being the end-all measurement and I don’t trust student surveys for an accurate portrayal of a teacher’s abilities.

Interestingly, right after finishing the Gates report I was given a link to this abstract from a paper issued in December of 2012 by C. Kirabo Jackson:

I present a model where students have cognitive and non-cognitive ability and a teacher’s effect on long-run outcomes is a combination of her effect on both ability types. Conditional on cognitive scores, an underlying noncognitive factor associated with student absences, suspensions, grades, and grade progression, is strongly correlated with long-run educational attainment, arrests, and earnings in survey data. In administrative data teachers have meaningful causal effects on both test-scores and this non-cognitive factor. Calculations indicate that teacher effects based on test scores alone fail to identify many excellent teachers, and may greatly understate the importance of teachers on adult outcomes.

I am interested in the idea of a teacher’s outcome on “non-cognitive ability”. How do you measure a teacher’s ability to help kids throw off the fear they all seem to exist in during adolescence as they work to figure out who they are and who they want to be? I don’t think a standardized test measures that well and there is something powerful about realizing you have the respect of an adult who is not your relative and who shows passion for life.

[UPDATE 01/17/2013]

Two new articles have come out that chime in about the statistics (or lack thereof) in the Gates Foundation article. Good reads. And if you teach stats they are rather applicable.

The 50 Million Dollar Lie, by Gary Rubinstein

Gates Foundation Wastes More Money Pushing VAM, by Gene V. Glass

Last night I got a missive from one of my darlings worried about her math final. If she doesn’t pass it, she’ll fail the class. This is her first semester in college. The note was short, and she ended it with

Why am I so bad at the maths?! D:

I wrote back the following:

For me, maths was more about accepting a different way of thinking than anything else and that can be the hard part. We’re not brought up in a logic-based society. So much of high school math is focused on procedural thinking–if A, then do B & C, and ta-da! And that’s not really math. That’s more like advanced baking. Useful, to be sure, but not a way of thinking. Since I don’t use those types of problems as my main push in classes it’s also why the typical A-maths kids don’t like me for a few months–they’ve never really had to think for understanding before.

And that right there is the key. Maths makes sense. It’s the Queen of Science for a reason. If you think what you are doing in math does not make sense or is magical, than we need to figure out a different way for you to think about it. Luckily, there are a lot of ways to think about maths that are successful. Unluckily, finding the way that works for your brain can be grueling.
I got through high school using procedural skill (the If A, then do B stuff). I hit a brick wall in my first maths class in college because it didn’t work with that professor. I had to show him how I was making sense of the maths and since I wasn’t really making any sense I didn’t have anything to give him. There were a lot of nights spent in the math study room and I didn’t really get the thinking needed to understand, but I was moving that way. Slowly.
Sophomore year was when it was finally clicking that I needed to build my own understanding so I could stop playing the memorization game (it wasn’t possible to pass some of my classes via pure-memorization as no ones memory is that good. And I was taking Latin at the time so my memory banks were full up with vocabulary).
So yeah, maths is hard if you’re taking the memorize ever permutation of a problem and how to solve it route. Figuring out the patterns behind the maths is a challenge, but it pays a lot of dividends. If you ever want to chat about your stuff, just let me know. I have skype and I use google hangouts quite a bit for my work. Just remember: I am 3 hours ahead of you

What I did in class wasn’t enough to get her to where she needed to be to be successful in college maths. I still focus too much on the procedural at times, but every year I moved more and more away from that as I built up skills toward a teaching style I never saw in high school maths. I’m curious to see how much my classroom skills will atrophe while I am out of one or if the level of ed-research, blogs, and consulting work I do will help me hold steady. One can only hope.

I miss my kids.

To take this survey, you need to be in the US and a 7-12th grade math teacher.

click here to go to the survey.

Thanks for your support in this. As a consumer of math ed research, I feel I should try to give back to it whenever I can.

Oh, and this is the tweet that spurred much of my ideas for the talk. Or rather, had me re-writing much of my ideas for the talk.

A Call to Action
IM&E Conference, Berkeley, Ca
October 14, 2012

I want to take everyone back in time to remember their early teaching years. Especially that first year. Was it crazy? Manic? Did you go to bed on Friday after getting home and not wake until Saturday? I remember the first time the fire alarm went off and I just stopped and stared at the blinking light in confusion.

Now that you’re back in that first year, I want you to think back to that first time the principal drops in and takes a seat.

How’s your heart rate doing with that memory?

Over the years you get used to it. I was fortunate enough to have an admin who I trusted completely to take the right side—that of the student—when we would have the sit-down meetings in the office in the days following the observation. Feedback is difficult, especially when it’s about a passion. Suggestions for growth are painful to listen to, but how else do we get better? And positive feedback from my Admin? To be told how you are doing good with the work you care about so much? Wondrous.

But that isn’t unique to teaching. That’s part of being a professional. Amateurs practice their craft where no one can see. Or hear. Or comment. Or help.

Teachers are not amateurs. But the system often treats them as such.

Those observations? Officially they were twice a year unless I asked specifically for someone to come in. Occasionally there would be 5-minute pop-ins. Most feedback I received on a daily basis was from students asking about meeting after school or before school or during lunch or for less homework or more time or nap day. And let us never forget the ever-popular ‘class outside!’ As a teacher from Seattle, that one still doesn’t make sense to me.

Day after day that is type of feedback that teachers hear the most. It’s very easy as a teacher to bury yourself in those voices. To wrap yourself in your classroom and devote all of your focus to your students. The work becomes entirely about the kids, safely contained in a room, with a door that is often closed. You can be a good teacher that way. But I think it’s very hard to do so.

The day I received the most feedback from my students I had changed one thing. Just one. I straightened my hair… Every period almost every student commented on it. It was crazy.

Yesterday I was having such a great time talking with everyone here and during lunch I sent out a tweet that said this:

Chris from Pennsylvania responded with this:

‘They must be willing.’ Admins, coaches, colleagues—all voices that can drown in the sea of student voices.

Timon from Canada said:

To which Chris responded:

How do you get teachers to see the value of professional engagement? My favorite comment came from Sadie, who is in Hawaii.

I believe many teachers don’t see the value in engaging as a professionals because they have no idea what that means. How do you value something you have never seen? Yesterday Donna said that CCSSM are a chance for teachers to “reclaim power as professionals.” She spoke of the potential of “[sharing] ideas across the miles.”

Today I want to show you what happens when teachers do stand up and claim their professional identity and how the online world is everything the staff lunch room should be. I would like to note that everything I am about to share with you was posted in the last week—I did not go mining through posts to find the best stuff. My goal in sharing this world with you is twofold: I want you to join it and add to the professional online world of education—sometimes referred to as the Edutwitterblogosphere—and I want you to show these things to the teachers in your sphere of influence. I want every teacher to understand the power of professional engagement and what it can mean for themselves and, most importantly, their students.

Let’s start with Fawn Nguyen.

This is her blog. Yes, bloggers like amusing titles. To me, this post is an example of blogging at its finest. Fawn has taken a video originally created as a 3-act by Dan Meyer, a former teacher and current grad student at Stanford. If you’ve not heard of 3-acts, give me a bit and I’ll share with you a place you can go and check them out. They are very cool.

This post details the entire first day of Fawn’s lesson. How she introduced the lesson, student dialogue, links to the worksheets she used, pictures of student work. At the end are her thoughts on what she plans to do tomorrow to bring the lesson to a close. My second favorite piece is the list of six reasons she believes this lesson worked for her students. And a thank you to Dan.

But don’t stop reading. General rules of the internet say to never, ever read the comments. That does not apply to teacher blogs. My favorite part of teacher blogs are often the comments. Fawn has exposed herself and her classroom to anyone who comes along to comment on. The first person to do so is Dan Meyer himself, thanking her for the report and phrasing her ratios approach as he was thinking Pythagorean Theorem. Thanking her for taking his work and doing some great with it. Other comments are from teachers showing appreciation for her work and her replies to them.

Fawn blogs regularly, and by that I mean 2-3 times per month, maybe more on occasion. If she’s getting comments on each of those posts from other teachers, think about what that means with respect to the student voices. She’s there for her kids, but she’s also listening to other education professionals and making her lessons better. Trying new things can be scary, but how many teachers could read something as thorough as her breakdown of that day and think ‘you know, what she did makes sense. I could do that with my kids.” And that’s how engagement starts.

This is Kate’s blog. She’s currently teaching in Argentina and taught in New York for many years. In the online teacher world, she’s kind of a big deal among high school math bloggers. She posted this on Friday. It’s not a lesson, it’s just something she did in her class. And idea. Using two laser pointers attached to a large board-compass, she’s drawn two green stars on either side of her whiteboard. Students have to hold the compass steady and figure out where in the classroom they can stand that puts both lasers on the starts. I’m not going to ruin the punch line, but if you go home from here and have access to lasers and you do not try this I am revoking your math geek card.

The Global Math Department is now in session and headed up by Megan Hayes-Golding of Georgia. She organizes individuals or small groups to present on a topic each week on Tuesday at 9 Easter, 6 Pacific. Last week Chris Robinson, Andrew Stadel, and Dan Meyer presented about the creation and utilization of 3-acts in the math classroom. If you create a log in for the site you can go back and watch old presentations. The week before was a variety of teachers talking about their favorite things. The week before that was on mathematics and art and featured a trio of teachers from St. Ann’s in New York City and how they started a math art class at their school and the math art day they started that the whole school participates in. Right before their part you can see me talking about my use of origami in the classroom over the years. Next Tuesday is Shawn Cornally of Iowa discussing standards based grading. You should check it out.

Tweeps—internet slang combining the words twitter and peeps—come from all over. This is a map made a while ago and doesn’t represent everyone, but it’s a nice way to see if there is anyone near you. If you look at the upper right green symbol on the map that is me. On the bit.ly link for the conference I’ll be sharing the link to this map and to a document that has a spreadsheet of educators, their twitter handles, their blogs, and a column of what they focus on. You can use that column to find, in example, other algebra 1 teachers to follow or K-5 specialists. There are not a lot of K-5 math folk on there, so I am hopeful some of you here today will consider putting your names and blogs on there.

180 Blogs are teachers taking a picture of their classrooms all 180 days of the school year and posting it along with a short info blurb. It’s mostly science folk. For now. This is a picture of the math classroom of Bowman Dickson, who teaches in Jordan. His students took a test today, so he posted what the classroom looked like yesterday. If you are wondering why his students were taking a test on a Sunday, it’s because his week is Sunday-Thursday, which the weekend Friday and Saturday. How could you use something like this in your own district? Do you know some teachers comfortable with exposing their practice that you could start something local—get teachers sharing across schools? Get teachers interested in what their colleagues are doing?

Ah, Pinterest. Stop laughing. For those of you unfamiliar, the word Pinterest comes from pin and interest. When you log in you get to create boards where you can pin things you find from around the internet that interest you. Yes, it is home to recipes and hair styles. It’s also home to a growing group of educators sharing neat classroom ideas they find around the web along with pictures of things they are doing in their own rooms. An former colleague of mine, Stephanie, has created boards for each type of class she teaches to remember all the ideas she comes across. I follow her on Pinterest, so I can look at her boards for ideas any time. Sharing things you find that enhance your practice? That’s being a professional.

MathMistakes.org is run by Michael Pershan. It’s relatively new, but already has about a hundred submissions. If you see an interesting mistake a student has made, you can snap a picture and send it to the website where it will get categorized both by topic and by CCSSM grade bands. The thing I love about this website is that anyone can comment on what they think the mistake was. Think about how pre-service teachers could use this. No one goes into math teaching with the innate ability to understand student mistakes. I remember many times in my early years where I would be grading and end up just staring at a student’s solution. I took to nudging more experienced math teachers and asking for help only to be amazed when they could glance at the page and tell me just what the student had done and recommend how to help the student’s understanding. It was like magic. What if departments used something like this? Just five minutes spent on an error that someone brings in? Can you imagine how great those five minutes of professional discourse would be at building a stronger department?

I could keep going with references. These are just a few of my current favorite things that came up on my radar this past week. I know of groups of administrators on twitter who have chats regularly. I know of teachers who do a book study with a different chapter each week they discuss in 140 characters or less. But I want you to think about the impact sites like these have on the people that create them or engage with them. Those student voices I talked about earlier? The ones that drown out everything else? They have a place in teaching. They need to be listened to. But they cannot be the only voice.

As Chris said, if we are going to help teachers create the best classroom experiences for their students that they can then that means we need to help teachers be willing to seek out others in the profession and engage at that level. Get them reading blogs of teachers just like them. Help them see that, yeah, trying new stuff can be daunting, but look at some of these payoffs. Look at all the footsteps for you to follow. Having someone I don’t know outside of that they are a teacher post on my blog that they loved my lesson and took my plan and tweaked it for their students and it went great is an amazing feeling. Knowing that other teachers out there are struggling with how to implement the CCSSM just like me and that they’re posting about what they are learning in trainings and reading their thoughts on the Standards for Mathematical Practice makes me feel like part of a group; not alone in my room.

So what are you going to do with what you have learned here? How are you going to help teachers become engaged members of the professional community? As Bill McCallum said yesterday, the “standards don’t implement themselves.”

And I want to emphasize that I believe Sadie is spot on when she names fear as the biggest hold back  Who can blame teachers in this age of value added and high stakes testing?

The web has given teachers the forum to publicly reclaim and elevate the profession of education. We have a space for all teachers to engage as professionals. Online I talk daily with teachers from around the country and sometimes beyond. Nothing I have accomplished has been done in isolation. Not my National Boards certification, not my student-nominated Teacher of the Year award from my school. My students call me Ms. Black. My friends call me Ashli. I blog at Learning to Fold. I tweet under the name @mythagon. I hope to see you online sometime. Come join the conversation.

Year 1 (2011)

• I had kids team up in groups of 3-4 and assigned them a specific graph with a specific section of the x-z plane to graph split into 4 blocks.
• Each student had to set their graphing window appropriately on their calculator.
• Each student had to make a template for their cards to ensure consistency between their cards and the cards of their group. Templates looked like this:

slot a card in and you can see the grid through it to help with sketching the graph. The numbers correspond to the x and y values of the students calculator window setting. A student came up with this idea.

• Since the TI-83/84 can only handle two variables, students would graph the equation multiple times using a different z-value each time.
• After each graph, they would copy their window onto an index card using the template. Do this enough times and you get a series of index cards you can line up in a row to see the shape of the plane:

Side view of a single students finished work inserted into the ‘rails’ of cardboard.

• Once all 4 group members have their blocks done, you slide them together to get the full view of the 3d graph:

The blue and orange cards in the back left are the same ones from the previous picture.

Year 2 (2012)

The colors correspond to different classes, in case you are wondering.

strongly recommend coloring then cutting and then inking the top bit for contrast

one student’s work placed into the foam board

you have to work from the middle outward. I tried to have the pieces overlap a bit to make them fit together well

window light makes the pretty

what you get if you stick the camera into one of the grooves

y = sin(x^2 + z^2)/(x^2 + z^2), in case you were wondering

Students in the district see a pinch of trig in Geometry and maybe a small bit in Algebra 2 if they took honors, but Precalc is the real beginning for them. I’ve scribd a teacher’s guide for the Unit Circle Project and a student worksheet that goes with it for anyone interested.

In other news, I am now living on the East Coast in Maine and working as a mathematics consultant for a few groups (teaching jobs are not to be had here due the remote nature of my new location). Hopefully this will mean I’ll get to clear out and post several half-done blog entries over the next while. Unpacking comes first, though. And if anyone is going to NCTM Hartford, let me know as I also plan on attending.

Don’t let anyone call you a ‘natural’. They may mean it as a compliment, but it diminishes the intellectual work that we do as professionals to become as good as we have to be to educate our children.

During our chat I was reminded of something I used to do that for some reason I have not been doing the past few years. I thought I would write it out here to help me remember to use it again in the future. I think best when I get a chance to write things down, so this may be a bit free-flowing.

So picture this: you have a lesson planned that will hopefully lead kids through some mathy ideas to a big conclusion. You are concerned, however, that they will not ask the questions you are hoping for and are unsure of your abilities to steer the conversation without obviously grabbing the wheel. This was my state of being for several years (and still is sometimes). On the spur of the moment in class during my 2nd year of teaching I decided to ask I kid I trusted to keep a straight face to ask a specific question if I gave a signal. It wasn’t critical to the question, but it was a nuance I didn’t think the kids were picking up on. I ended up giving the student the signal and they asked the question. This caused a pause in the class and then more back and forth conversations in the group about this point. My plant jumped into the discussion and none of the students thought anything out of the ordinary had happened.

Was I covering for weak group-discussion-leading skills? Maybe. Did the kids see something they wouldn’t have seen otherwise had the question not been asked? Yup. Was it better coming from a student than it was from me? I think so. I believe that students often respond better to the questions/responses of their peers than they do to myself. Peanuts effect and whatnot.

Fast forward a bit. I ended up using plants on occasion. Sometimes for questions, sometimes to say wrong answers I wanted to make sure got covered. I totally got caught in some classes, which I was able to play up enough that the kids found it amusing and the plant was giving off smug ‘chosen one’ vibes. I realized that getting caught could be great. I chose to sometimes give a kid a written question/comment with vocabulary they would never use, but I totally would. Authentic learning environment? Nope. Did kids pay attention, get a laugh, and build not only mathematical understanding but also classroom community? yup.

I once used it freak out students. Used to be my 1st period support kids would see me twice: once for support and once for regular algebra 1. I had a kid ask in support about other math symbols (we were doing inequalities if I remember right). This was a student who expressed dislike of math, but was well-liked by his peers and rather good at math he ‘got’. I wrote out some abstract form of “for all x contained within the reals ….” in symbolic notation. he thought the ‘code’ was pretty cool, so I asked if he could remember what it meant. He repeated it back. Fast-forward to period 2. Regular algebra 1. Same symbols question comes up. I get my plant a quick glance and he is playing it cool. I write up the symbols again, tell the class it’s an advanced math sentence and ask if any of them know what it says. They are, of course, stuck on the upside-down A. My plant raises his hand and TOTALLY plays this up. Squints his eyes a bit, rubs his chin, “well, I think it says …” Beautiful performance. I wanted to applaud. Every other kid in the class is staring at him, his friends are whispering demands for how he knew that. I congratulated him and repeated what he said while pointing at the sentence and then continued with the lesson.

I’m pretty sure he admitted to them later about being a plant, but it definitely got the attention of the class and I started seeing the upside-down A and the triple dots for ‘therefore’ on some papers.

Some notes on the details. I never used the same kid twice. I never made it a regular thing lest they get paranoid about one another. I’ve not done it in years. I think in the National Board year-of-crazy it got lost and I hadn’t even thought about it (which tells you how infrequently I did use it).

I’m much better at helping to direct conversations and making sure that kids are set up properly to make the connections I want these days. That does not, however, mean I am not thinking about how to use the ‘plant’ idea in some of my upcoming lessons.