This year, as I have been in the past few years, I’ve been attempting to incorporate more writing in my math classes [note: Shelli found a post from 2009 I wrote on this endeavor]. It’s been extraordinarily enlightening, because what this has done is show me two things: (1) kids don’t know how to explain their reasoning in clear ways, and (2) I’m usually extraordinarily wrong when I think my kids understand something, and the extent to which I am wrong makes me cringe.

(wow, been too busy to shave, have we Mr. Shah?)

For the first point, I don’t actually do much. I ask them to write, they write, I comment. And we discuss (more at the start of the year, but I always let this go and I forget to talk about it a lot). In Algebra II, they get one or two writing questions on every assessment. And each quarter they had problem sets where they had to write out their thought processes/solutions comprehensively and clearly. Even though I didn’t actually do anything systematic and formal in terms of teaching them to write (mainly I just had them write), I can say that I’ve seen a huge huge improvement in their explanatory skills from the beginning of the year. What I used to get just didn’t make sense, honestly. A random string of words that made sense in their heads, but not to anyone reading them. But now I get much more comprehensive explanations, which usually include words, diagrams, graphs, examples. They aren’t usually amazing, but they’re not ready to be amazing.

For the second point, I realized that the types of questions that we tend to ask (you know, those more routine questions that all textbooks ask) don’t always let me know if a student understands what they’re doing. It just lets me know they can do a procedure. So, for example, if I asked students to graph , I would bet my Algebra II kids would be able to. But if I showed them the question and the solution, and ask them to explain what the solution to that question means, I would expect that only half or two thirds of the class would get it right. (Hint: The solution is the set of all points (x,y) which make the inequality a true statement.) They can do the procedure, but they don’t know what the solution means? That’s what I’ve found. And you know what? Before asking students to write in the classroom, I had deceived myself into conflating students being able to answer with a full understanding of 2-D linear inequalities. [1]

Before having students write, I actually believed that if I asked that question (“What does this solution mean?”), almost all the students would be able to answer it. (“Like, duh, of course they can!”) But since asking students to explain themselves, explain mathematics, I’ve uncovered the nasty underbelly to what students truly understand. The horror! The horror! But now that I recognize this seedy underworld of misconceptions or no-conceptions, I’ve finally been able to get beyond the despair that I originally had. Because now I know I have a place to work from.

The counterside to this point is that when kids do understand something, they kill it.

This simple question I made for my calculus students early in the year, and this student response, says it all. I have no concern about this kid understanding relative maxs and mins. No traditional question would have let me see how well this student knew what was up.

For me the obvious corollary is that: we need to start rethinking what our assessments ought to look like. If we want kids to truly understand concepts deeply, why don’t we actually make assessments that require students to demonstrate deep understanding of concepts? I am coming to the realization that the more we keep giving the same-old-same-old-assessments, the more we are reinforcing the message (implicitly) that we don’t reallyreally care to know about their thinking. We are telling our kids (implicitly) that we are content if they show their algebraic steps. But as I’ve noted, my big realization is that students performing those algebraic steps don’t necessarily mean that the student knows what they’re doing, or what the big picture is.

I don’t know have an example of what I think a truly ideal assessment might look like, but I do know it isn’t anything like I gave when I started off teaching five years ago (has it really been five years? why am I not better at this?), and I do know that each year I am slowly inching towards something better. Right now, my assessments are fairly traditional, but with each year, they are getting less so.

Sorry if I’ve posted something like this before. I have a feeling I have. But it’s what’s been going through my head recently, and I wanted to get it out there before I lost it.

[1] Another good illustration might be having students solve . Sure, they can get . But does doing that really mean they understand that whole “if you divide by a negative in an inequality, you switch the direction of the inequality” rule that has been pounded in them since seventh grade? Nope. The traditional questions don’t tend to check if the kids know why they’re doing what they’re doing.

I participated in a great twitter conversation the other day where we brainstormed a few strategies to help make our courses more accessible to English Language Learners (we used the hashtag #ELLmath, the approximate transcript is here if you are interested). It was a great start to what needs to be a running dialogue for me, as I teach almost 100% students for whom English is not their first language. If anyone has any ideas about #ELLmath, I would love to hear them in the comments. The conversation reminded me of a little idea I had last year, playing the game Math Taboo to help students expand “definitions” to actual understandings of concepts. Now, I’m sure other people do this, and a quick Google search leads me to believe it’s not all that novel, but while discussing #ELLmath, it struck me as a particularly good exercise for ELL students.

The idea of the real game is to get your partner to guess a word by describing without using any of the five taboo words, which are usually the first words that anyone would go to in a description. So the obvious math equivalent is to pick a term that you are throwing around in your class and get students to describe it without using their go-to math descriptors.

We played during beginning-of-the-year-review as a class, with the word to guess already known to everyone, and I gave students a chance to take a stab at verbalizing a definition without using the taboo words, one at a time until we got an acceptable description. However, this could easily be adapted to be a much more interactive activity (though its creation might take just a bit of time).

So why play this?

Whenever working one on one with students, I found myself trying to diagnose why they were not understanding a problem. I would ask them things like, “Well, what is a derivative anyway?” and they would often answer with something that I found acceptable, but perhaps could have been just something that they had figured out should be said as the “correct” answer. Even if they weren’t saying book definitions (which would actually be easier to deal with), many times they were using my informal definitions – words that they had internalized about the concept that might not actually display a deep understanding, but that I had been mistakenly accepting as evidence of learning. Definitions are important, but assuming that those are indicators of deep understanding is, of course, very problematic, no matter where those definitions come from.

So, this Taboo game serves a two-fold purpose: learning for the students (by forcing them to think deeply about a mathematical concept; by having them trade in math jargon for conceptual understanding; and by hearing classmates describe something in more accessible vernacular) and learning for me (by seeing how well students actually understand a concept; and by seeing what language students use to talk math in the hopes that my mathematical narrative can better reflect theirs in the future).

Alternative game: In how few words can you express this definition?

I have never tried this game I’m about to describe, but the idea is to start out with a long definition from a math textbook and see how few words you can use to express the same idea. Delving into the Twitter world this summer I have realized how wordy I am, and the process of editing my tweets down has made me realize how many words I use that are unnecessary. Twitter forces me to think about what is the core of my idea, which led me to think up this exercise. This could be done competitively (give groups 5 minutes to brainstorm), or you could do it countdown style, trying to lower the number of words by one each time. This could get students to really consider what is important about a mathematical concept and to get them to realize that the thing itself is more important the words you use to express it.

Last year I had archived 2,270 emails to and from (and about) students. (I make a separate folder for each student and file everything related to that student.) The breakdown from the data last year was here.

This year, the total comes to 3,364 emails. The reason for the increase is probably due to the reassessments in calculus… students had to send me an email justifying their request for a reassessment.

The student with whom I had the least communication to/from/about had 16 emails. The student with the most had 192 emails. Each student on average has 71 emails.

Drumrollllll please…

Here are my 2010 Edublog Award Nominations…  (my 2009 nominations are here)

Best individual blog: Shawn Cornally’s Think Thank Thunk

Shawn’s blog arrived on the scene in Februrary 2010, and has quickly won over the math teacher blog community. He’s a staple read. Why? Clearly he’s a compelling writer, with posts chock full of quotations inspiring enough to lift and emblazon on classroom banners. But it’s more than that — than the inspiration dripping from every post for the readers to hold out our cupped hands and collect. It’s that Shawn arrived on the scene with a simple philosophy about the classroom, and built a whole curricular and assessment-based regime around it. The philosophy is this: LET KIDS LEARN! GIVE THEM FREEDOM AND THEY WILL LEARN TO THRIVE. This philosophy lends itself perfectly to Standards Based Grading, and his work, coupled with Dan Meyer, Matt Townsley, and others, have led to a wholesale shift in the math edublog community. At the very least, spearheaded by Shawn, Standards Based Grading led to serious, questioning, and respectful conversations around the nature of assessments. And for some of us, it has led to a wholesale reorientation of how we view the classroom and how we think about learning. Did I mention that Shawn only started blogging in February 2010? His blog is a Tour de Force.

Best individual tweeter: David Cox / @dcox21

Reason: He’s always there, and his snarkiness is always on, ready to go! If you need any convincing, check out any of these “Favorite Tweet” posts. He also tweets about teaching.

Best new blog: Shawn Cornally’s Think Thank Thunk

Reasons: listed above

Best resource sharing blog: Mimi’s I Hope This Old Train Breaks Down

Reasons:  I don’t think Mimi gets the recognition she deserves. She has created countless amazing resources for her classroom, and shares them. These worksheets and activities are inspiring enough that… well, let me put it this way… she teaches Geometry, and I dread the day that I have to teach Geometry, but knowing I have Mimi’s stuff at hand makes me think teaching it could almost be fun.

Best teacher blog: Kate Nowak’s f(t)

Reasons: Kate is our Fairy Blogmother. Her blog doesn’t have a singlular focus, but that’s part of its charm. Kate blogs about whatever, and not knowing what’s coming next is part of the appeal. One post might be on special right triangles in dollar bills, and the next her trying to work through the muck and mire of Standards Based Grading. Kate also spends time drawing new people into the blogging community, and she even wrote a post on how to start a math teacher blog that I dare say spawned quite a few new faces in our expanding circle. Her blog is a perennial staple, read and admired by all.

Best use of a PLN: Riley Lark and his Conference on Soft Skills

Reason: Riley Lark created what I think is the first Virtual Conference for the online math teacher community. In general most of the posts that we write, and read, deal with curriculum — and how we get our kids to know it, and how we know they know it. Riley saw a gap in the conversations. We don’t talk about those other things we do as teachers to be effective: those things we do to connect with students so they know we’re there for them, and we want them to be there for us. The relationship building that goes on in the classroom… we all do it tacitly. He asked us to make explicit what we do implicitly. This conference started with 5 “speakers” and blossomed into 17 “speakers,” each writing with our own voice. Riley called upon our PLN and our PLN responded.

Lifetime achievement: Dan Meyer’s dy/dan

Reason: He’s part of the first generation of math teacher bloggers. His cause celebres have changed, but his overarching goal is the same: to get others to think about ways to engage students in the classroom. From focusing on design, to focusing on assessing, to focusing on his What Can You Do With This (WCYDWT) series, to focusing on pseudocontext, Dan has asked provocative questions since 2006. He asks teachers to re-look at what we’re teaching, and asks us to pique the curiosity of our students. Specifically, his mantra (though he has never put it this way) is don’t put the cart before the horse. Don’t teach the concept and the method of solution, without motivating the need for the concept. Grab your students’s curiosity by showing them they need a concept, so they want to learn the concept. And let me tell you, it is easier said than done… but Dan makes it look effortless. He has inspired the second wave of math teacher bloggers, which have since inspired the third wave. Yes, in online terms, 2006 – present is a lifetime.

My Algebra II kids don’t like to read the textbook. Heck, neither do my calculus students. This isn’t surprising. It’s extra work and it’s hard. My class also makes it hard for them, because I do not use the textbook as a skeletal structure for the course. I teach mainly out of my own materials, and use the book more as a supplement.

But that doesn’t mean that I don’t want them reading math. Kids are never taught to “read” a math textbook. If they ever do approach a math textbook, they approach it like a history book. The read it linearly. They also read it passively. Their eyes glaze over. They read words, but they don’t try to connect the words to the equations or pictures. They don’t read with a pencil in their hands. They hope for some Divine Knowledge to descend upon them simply by having the book open and their eyes on it.

That doesn’t work. We all know this. Reading math is an active thing.

And so recently I’ve started talking with my class about it. To start this process/discussion, one that I hope continues, I gave my students a worksheet to fill out (see above). I love the honesty with which they responded.

For question A, some representative responses:

“I read what was assigned to me but did not read anything extra.”
“I find that textbook reading is pretty boring, so I don’t do it unless I have to.”
“I did not because I had assumed I wouldn’t learn things I needed. All I would do was look at examples.”
“No, I find it difficult to understand math when reading it in paragraphs; it makes more sense to me with a teacher.”
“I did not generally read my math textbooks. I did, however, always look over the example problems.”

Some responses for Question B:

1. The writing can be confusing, wordy, and not thorough
2. The book is BORING
3.  Small print
4. Too many words for math
5. Outdated examples

Some responses for Question C

1. Everything is all in one place
2. Have a glossary
3. Can read at own pace; refer pack to the text when I get stuck
4. Sidenotes! Diagrams! Pictures
5. Real life examples
6. Definitions clear
7. Key terms are highlighted
8. Wide range of example problems with step by step instructions
9. Colors!

I hope to do more as we go along. I might have them learn on their own, using the textbook (and the online video help) a whole section or two. There’s no reason they can’t learn to use the book to be independent learners. I will give them class time and photocopies of the section they need to learn, and they will have to figure things out by the end of the class for a 3 question quiz.

I also hope that by the end of the year, we can use their critique of math textbooks for them to write their own textbook. Okay, okay, not quite. That’s way too ambitious for me. Two years ago I had my Algebra II kids write really comprehensive Study Guides for the final exam. This year I might ask my kids to pick some of the hardest material and create their own “textbook” for it. They’ll get to write it in pairs, and then they can share their finished product with the rest of the class. That will probably happen in the 3rd for 4th quarter.

Anyway, I thought I’d share. Since I like to emphasize the importance of mathematical communication to my kids (though I don’t do it nearly enough), I thought I’d talk about this one additional component in addition to getting students to talk and write math… READING MATH!

Kate “I’m not snarky” Nowak and Sean “the squirm” Sweeney are working on putting together a…

“Welcome to the internet, math teachers”

page on the internet. As much as I like them, I feel like this project needs a new working title. Because that’s just sad.  In the meanwhile, you should fill out their survey here.

Full confession: I haven’t filled it out yet. But I’m going to, today. So there. THERE. I PUT YOU IN YOUR PLACE FOR CALLING ME A HYPOCRITE.