Grand Challenges for the MTBOS

Earlier this year NCTM asked folks to fill out a survey to help identify "Grand Challenges in Math Education." This was sort of funny (a survey?) but also sort of fun. I found a bunch of blogged responses interesting. (this this this this)

Anyway: I like this game. Let's play it for math teachers on the internet!

What are grand challenges facing math education that math teachers on the internet might set their eyes on? These challenges should be neither too narrow ("Create a central repository for AP Stats worksheets") or too huge ("Change the way math teachers across the country teach"). A grand challenge should have an impact on the profession beyond the internet.

Fun game, right? OK I go first.

A Grand Challenge for the Math Teacher Internet

Discover, describe and validate paths toward improving teaching beyond the conventional paths toward teacher-improvement.

If the question is "How do teachers get good at teaching?", the answer is typically "Ed-school, professional development and departmental collaboration." While the conventional paths can be great, they are basically a no-go for lots of teachers. Teacher-prep programs are of wildly uneven quality and teachers face a variety of constraints in their own schools.

How can teachers get good at teaching on our own initiative? Without supportive colleagues or mentorships or high-quality professional training?

One thing that's true about math teachers on the internet is that they're motivated to improve. That's often why they're online in the first place. This opens up a whole host of other questions: What sort of teacher activities make a difference for one's teaching? What activities don't? Does talking about teaching on twitter help one's teaching? Does blogging? There are different styles of blogging -- are some more likely to lead to classroom improvement than others?

This is a big and juicy question, one fit for a grand challenge. It's also one that the math teacher internet could tackle. Blogs and tweets leave a record that can be analyzed for signs of improvement or changed thinking about teaching. Grad students and researchers could partner with teachers to analyze instruction and suggest directions. It could unify a lot of the work that goes on toward professional learning on the internet, uniting TMC, the Global Math Department, twitter chats and blogs in one common effort.

(It's a fantasy, sure, but that's what the game's all about, right?)

Questions:

1. Forgetting its feasibility, is this a worthwhile goal?
2. What's your grand challenge?

CGI is for High School Teachers

There's a new edition of Children's Mathematics coming out soon, and I got my paws on an advance copy of the book. I like the people at Heinemann (the publisher) and I like the book so I said I'd write a few words about this new edition, and the book more generally.

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Here's my story: I'm a high school teacher that got a job at a school where the high school teachers also teach little kids. Last year I taught 4th, 5th, 9th and 11th graders. This year it's 3rd, 4th, 5th and 9th. I love it.

I came across Cognitively Guided Instruction (the research program) and Children's Mathematics (the book describing it for teachers) when I started teaching elementary school students. I knew that this was new territory to me, so I asked a very knowledgeable friend to help me out. He sent me a reading list, and CGI was at the top of it.

CGI is fundamentally about arithmetic. It's about how kids learn arithmetic, how their strategies usually (always?) develop, and it's about taking all of the knowledge that teachers have about student thinking and mushing it into a useful system. It's this theory of strategies that makes the whole CGI program so valuable. Anticipating a kid's thinking is the closest thing we have to a teaching superpower, and CGI provides a system of anticipations for the classroom teacher.

One thing the first edition of the book was not about was activities or tasks. Unlike most of the math education books I read, there were no sample lessons or detailing of pedagogical moves. It's really about how kids think about arithmetic. From the 1st Edition Foreword:

"During this conversation, I realized how serious [the researchers] were about respecting teachers' judgments on particular issues. Since they had little evidence about representing these situations, they would see how teachers and children handled it. As they worked with teachers, sharing their research knowledge about students' learning of addition and subtraction, they would continue to learn from teachers and children."

I loved this focus on student and teacher thinking. The constraints of teaching are really so different in different environments. Attempts to change the way teachers act seems misguided -- much better to improve the way we think.

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Enough talky-talk. Here are two lists. The first is a list of reasons why I think high school teachers should check out Children's Mathematics, and the second list describes some differences between the first and second editions.

Reasons Why CGI Is For High School Teachers:

• Problem Solving: CGI is a study in how students think about arithmetic, and it's this sort of deep understanding of student thinking that enables teachers of arithmetic to drive student learning through problem solving. If you're interested in problem solving at high school, here's a model of the sort of knowledge needed to pull that off successfully.
• Modeling: There's a chapter titled "Problem Solving as Modeling." Cool idea, right?
• The High School Disadvantage: Reading CGI (and then teaching elementary school) made it clear to me why students have an easier time learning arithmetic than they do learning high school math: time. Learning takes time, and high school teachers are at a huge disadvantage because of the ridiculous flurry of topics that need to be taught. Watching student thinking develop through these pages drives that point.
• We Need More Research: CGI offers a systematic overview of how students learn arithmetic operations. Nothing like this exists for most high school topics and it's a shame. What's the development that kids pass through while they're learning quadratics? Complex numbers? Exponents? We just don't know, and it would be amazing if we did.

Differences Between the First and Second Editions:

• The new text is full of links to videos. I was hugely impressed by the quality of the videos. The camera never strays from the kid. A question is asked, and you watch the kid's reaction. One by one, you watch kids use the strategies detailed in the book.
• The first edition didn't really dwell on how and when students might think in writing, but the new edition does a nice job with this.
• The authors are far less cautious about offering classroom recommendations in this new edition than they were in the first. There are two new chapters detailing their advice for teaching through problem solving in elementary classrooms. I'm sure that many will find these chapters helpful, but there's something austere and lovely that I'll miss in the restraint of the first edition.
• Stray observation: they bumped up the magnitude of the numbers in a lot of the examples. Interesting!
• After each chapter there are a whole series of exercises for working through the ideas of the chapter. I was skeptical, but after working through the exercises from one chapter I was impressed by their quality. CGI is accessible, but it's still an interconnected system of strategies and patterns of thought and it took me time to get down. The exercises helped.

It's a good book, and it's definitely worth 27 bucks. Get it for your birthday.

Feedback Roundup, Vol I

Effective feedback continues to be a preoccupation of this blog. Here are some posts from other teachers who are thinking hard about feedback and related issues.

• Do you just sit down with a stack of papers and a red pen and let loose? Mary Dooms suggests that a certain amount of planning needs to precede the actual writing of feedback. She recommends looking through the entire class set of work before committing pen to paper. (link)
• John Burk gave a student some detailed written feedback, and on twitter he wondered whether it was effective. Follow the twitter conversation for John's thoughts about effective feedback, "metacognitive" lessons and giving his students a grade on how they respond to feedback. (blog, twitter)
• "Just because I'm giving multiple choice tests, doesn't mean I have to give binary feedback." The hard part, though, is figuring out what sort of feedback to give! Justin Aion offers encouragement and hints next to wrong answers, and then gives his students time to improve their earlier work. He's disappointed that his kids don't use this time well, though, and he's leaves with a lot of questions about how to make feedback work in his classes. (link)
• Learning software aims to give immediate feedback that serves a very different purpose than the delayed feedback (that has shown to be perfectly effective and) that teachers often give. Dan Meyer shares a piece of software that aims to help make delayed-feedback and little bit more immediate. Check out the post for links to research on feedback and reviews of the software. (link)

This is the first in a series. If you have a post on feedback that you like, please share it with me either in the comments or on twitter.

Two Estimation Tasks

Here are two estimation problems.

The first is from TERC Investigations:

The second is from Andrew Stadel's Estimation180:

I have some questions about the relationship of these two problems to each other:

1. Do these two problems help students get better at the same thing? Or do these two problems help students improve at different skills?
2. What sort of knowledge is needed to successfully tackle the first estimation task? The second?
3. What does the first problem seem to mean by "number sense"? What does the "second"?
4. Can we infer why estimation is valuable to the writers of the first problem? The second?

Thoughts?

How I'm Trying To Mentor Myself

Reflecting on teaching helps make teaching better, right? Blogging about teaching is a form of reflection on teaching ergo blogging makes you a better teacher.

The pic at the top of this post comes from Dan Meyer's presentation about teachers who blog or tweet and what they get out of said blogging/tweeting. I helped him a bit with the research for this talk, and the "blogging as reflection" idea came through again and again in the responses to Dan's surveys. Blogging is reflection. Reflection leads to better teaching. Better teaching? More blogging. Boom. A virtuous cycle.

I'm sure that blogging is reflection. But I don't think enough teachers grapple with the limits of blogging as reflection. And there are some pretty serious limitations.

• It's very difficult to talk about specific incidents or students in a blog post. Readers lack the (boring) context of the classroom that could make those incidents meaningful.
• The vast majority of people are unwilling to write publicly about their anxieties and concerns about their teaching, so the vast majority of posts are about sharing moments that worked.
• Blogging pushes us towards reflections that can easily cross the threshold into other people's classrooms. I'm more likely to blog about "How does feedback work?" than I am about "How can I make sure that my fourth graders each have strategies for figuring out a multiplication problem by the end of the week?"

I love blogging (obviously) but lately, I've been trying something different for my reflecting. I've been calling it self-mentoring, which is good except it's a terrible name.

A brief aside: as a new teacher, my school assigned me a mentor, and it was the best learning experience about teaching that I've had so far. My mentor visited my class each week. Before she observed, we'd chat for about 30 minutes about how my week of teaching had been. She'd ask questions and push me to think more deeply about why my students were doing, thinking and saying the things that they were. Post-observation, we'd have a similar chat that was laser-focused on what had just happened in class.

I haven't worked with a mentor since that first year, but this summer I found myself wanting very badly to recapture those conversations. Truth be told, this is something that I'd been trying hard and failing to do ever since our mentorship ended. (See here here here here and here for all my public failures.) The failures were easy to come by, because the problem is fundamentally hard: how do you push your own thinking past the limits of your current thinking?

I have 30 minutes free on Wednesday afternoons, and I like the way that I've spent them over the past few weeks. I seat myself and lay out my special teaching notebook. I have a protocol for how I spend this time. Right now, it goes like this:

1. Write about anything of interest for a moment or two.
2. Make a list of classroom incidents that might be interesting to dig deeper into.
3. Write down some questions.
4. Start writing about one of the classroom incidents. I have my mentor's voice in my mind as I do this. I try to ask myself the questions she'd ask. "Why do you think that happened?" "What do you think she was worried about?" etc. 
5. I make a little box for any significant takeaways from the analysis.
6. I go read through the Teaching Works practices and spend a moment noting anything related to those 19 practices that might be worth working on over the next week.

I've been really happy with the way this has been going so far. It feels realer and more natural than any of my other failed experiments over the past four years. I've been coming out with a bunch of takeaways that would make for terrible blog posts: "I sometimes walk away from a kid before I understand their thinking, and this makes it hard to decide who to choose to share in discussions" and things of that sort.

It's a lovely little ritual. I hope that I stick with it, but I think that I will because it's a lot of fun and it's been helping me make improvements. I'm hopeful that I've landed on something that works for me.

Do kids need help learning how to ask interesting questions? (Continued)

"Let's call out the elephant in the room: we're in math class. Chances are extremely high that mathematical questions are preferred. I think this should be discussed with students AND modeled by the teacher." (-Andrew)

"I think kids think it is reasonable to focus on math in math class, especially if they believe us when we tell them it's useful and lets us find out cool things." (-Julie)

"So when we ask for questions, we honor all the questions. We're grateful for all the questions. Then we model how mathematicians think, how mathematicians ask questions, and we ask our OWN questions that we know lead towards productive mathematical goals." (-Dan)

I was grateful for the comments on this earlier post. It seems to me that while many of you agree that mathematical questioning ought to be taught, we're caught between two different teaching strategies.

Strategy 1: Implicit Instruction

Honor every question or curiosity. Get excited by what the kids are excited about. But then always pick up on the questions that are mathematically productive. Say things like "We'll try to answer as many of these as we can." and "I love all these questions, but today we're going to focus on..." Say these on many days.

Show that you're excited and curious about your own question. Show that you have questions that animate and excite you too, and they just happen to take us to interesting mathematical places.

The kids will pick up on this, if not at first then with time. They'll want their questions to lead to the interesting mathematical discussions. If only mathematically productive questions lead to prime-time, they'll learn to ask mathematically productive questions. They'll be standing at the edge of a culture, peering in and trying to fit-in. It'll take time, but it'll stick.

Strategy 2: Implicit Instruction + Explicit Instruction

Help kids see things that they'd struggle to notice. Speed along their learning process by pointing out aspects of questions that make them mathematically purposeful. Say things like "What interesting mathematical questions could we ask?" or "I love questions that aren't just a matter of opinion."

Honor questions, yes, but also use praise, questioning and advice to help guide students towards productive lines of questioning: "I love 'How many?' questions."

Asking mathematically productive questions is hard. If we rely on implicit instruction then maybe some of our students will learn to ask great questions, but it's unlikely that all will. And if we want equitable outcomes, shouldn't we care about helping all students get there?

Explicit instruction can be done well and need not involve being dismissive of a student's natural curiosity. We use explicit instruction to teach all sorts of concepts, skills, strategies and practices. Why not use it to help students ask productive questions as well?

Questions:

1. When is implicit instruction preferred over a mix of implicit and explicit strategies? Why?
2. Is "modeling" a form of "implicit instruction"? Are they synonyms? Are there are other forms of implicit instruction?
3. The concern with explicit instruction seems to be that any explicit instruction on questioning would necessarily show that natural curiosity isn't valued by the teacher. But implicit teaching techniques aim to alter student questioning, anyway. Is the difference whether students notice that they're being guided? If not, is there some other difference?
4. Is there necessarily a trade-off between equity and discovery in teaching?

Written Feedback: Four Cases

Four quick summaries of the problems that students worked on, and the feedback that I gave each group the next day. A snapshot of the group's product before and after feedback.

Case 1: Handshakes and Diagonals

1. How many handshakes would it take for us to all shake hands?
2. How many diagonals does a square have? A pentagon? A hexagon? Find a relationship between the number of sides and the number of diagonals.

Case 2: Grids and Squares

How many squares can you find on this square grid? How about on this array of dots?

Case 3: Nets of Cubes

How many nets does a cube have?

Case 4: Cross-sections of Cubes

Here's a list of shapes. Can these shapes be the cross-section of a cube?

Questions:

1. Were all of these pieces of feedback equally effective, based on the whiteboards alone?
2. Each of these pieces of feedback were printed out and handed to a group. Would oral feedback have been better here?
3. Each of these pieces of feedback ended with a question. Would a direct suggestion have been more effective here?