# Rational Expressions

Hey, guys, it's Michael Pershan's blog!

## Sunday, October 19, 2014

## Sunday, October 12, 2014

### Thinking Through a Decision: Tom and Addition

**The Class, The Kid, The Activity:**

**I'm going to share with you a situation from my 3rd Grade class and a decision that I made. Then, I'm going to ask you what you'd do in this situation.**

This post comes with a question and a meta-question. The question is, "What's the right teaching decision?" The meta-question is, "How much context needs to be shared so that we can ask useful teaching questions?"

I'll attempt to provide context at three levels: about the class that I'm teaching, about the kid that I want to focus on, and about his work on a particular activity.*

** I'm unsure how important school context is to this. Briefly: I currently teach at Saint Ann's School, a private school with competitive admissions in New York City.*

**The Class: 3rd Grade Math**

**My 3rd Grade class has 11 students. My school has about eighty 3rd Grade students, and about half of them are in a "fast-paced" track and half of them are in a "regular-paced" track. This class is a regular-paced class. We meet less often that most math classes meet: 4 times a week, 35 minutes a day.**

We've been studying addition for the first five weeks of school as part of our first major unit. (I have lots and lots of curricular freedom, but I've elected to mostly use the TERC

*Investigations*activities so far.)

Here's a sample of some of the work that we've been doing in class over the past few weeks.

**The Kid: Tom**

**Tom is a 3rd Grader. (Tom is not his real name, but privacy etc.) Tom has worked slower than other students in class on most problems that we've done this year. I randomly call on students fairly often, and when Tom's name gets picked he often struggles to articulate a line of reasoning.**

On independent work, Tom works slowly but carefully. He often articulates his reasoning clearly using equations. Here's a sample of Tom's independent work.

Tom has had moments of great thoughtfulness over the first few weeks of school, and it's been fun to watch him make progress. In sharing his reasoning he sometimes makes mistakes ("65 cents and 25 cents makes a dollar") but when given a chance to slowly think it through he's able to improve his thinking ("No that can't be right because 65 and 20 make 85 and 85 and 5 make 10...")

Another nice moment from Tom: he solved 27 + 8 mentally by breaking 8 into 3 + 5 and "making a 10" with 27 and 3.

In short: Tom uses a variety of strategies quite well, but his addition is inefficient and error-prone.

**The Activity: Formative Assessment for Addition**

After a few days of some rich and interesting number talks, I wanted to know what addition facts my students still struggled with so that I could properly steer things in the next week. I knew that a few of my kids, including Tom, still struggled with mental addition, but I didn't have a good understanding of the types of problems that they struggled with. To assess which addition facts my kids "just knew" and which they had to figure out, and also whether they could figure them out, I decided to give my class 33 addition questions and ask them to work on them alone.

Knowing that these sorts of activities sometimes cause anxiety in students, I was careful in how I framed the activity for kids. I didn't tell them that the activity was timed. I took care to make sure that there was another activity ready for students who chose to move quickly that they could silently access. I also made it clear that I expected there to be questions that they didn't "just know."

Here's a video of how I introduced the activity.

Here's Tom's work on this activity.

*(About the stars: instructions to the class were to "start with the ones that you 'just know' and put a star next to any that you need to think more carefully about, and then work on those. By the time I noticed that Tom wasn't returning to the ones he skipped, I decided that this would be helpful information anyway and there was no need to have him work on these problems. That might have been a mistake.)*

I noticed that Tom has very little problem with adding 9's. Knowing his sometimes-facility with certain addition strategies (and his lack of quick recall of addition facts in general) I think that Tom is reasoning relationally here, e.g. 9 + 7 = 10 + 7 - 1. This is nice.

I noticed that Tom skipped lots of questions that involve adding 8 or 7. I noticed that he didn't "just know" 3 + 7, which might have been a useful fact for figuring out 7 + 6 or a lot of the other 7's.

Tom's work wasn't typical of the class'. Most students slowly, but accurately, answered all the questions. Casie's work (below) was more typical of the class. (The rest of the class' work is here.)

**The Question and a Decision:**

I've had students like Tom in every class I've ever taught. How do I help Tom while still pushing the rest of the class forward?

After thinking things through, here's what I decided:

- It seems to me that Tom could use some more strategies. In particular, there's a handful of additions that he could know very quickly if he started seeing 6+7 and 8+7 as "near doubles."
- Tom could also answer a lot of the problems that he missed using a "make 10" strategy. But in whole-class situations he's already used this strategy. Why didn't he use it on this assessment? He might just not have made the connection, or he might not "just know" the numbers that combine to make 10. (He missed 3+7 on the assessment.) I also think that "make 10" requires you to hold a bunch of numbers in your head, and that this strategy tends to be more cognitively demanding than other strategies. He might not be ready for it yet.
- Tom needs a chance to rethink the questions that he didn't get and make it an explicit goal to improve on them. Here's how I'll do it: I'll do a number talk that makes some of the useful strategies explicit, then I'll return their assessment work and ask them to work on any of the questions that they skipped or found difficult. I'll suggest that they use some of the strategies that were just discussed. (An issue is that many kids won't have any problems to work on, and I don't want to draw a distinction between those who
*do*and those who*don't.*I'll mitigate this by giving everyone their work to improve and a follow up assignment that they'll work on individually and silently.) - Over the next few weeks I need to give Tom an opportunity to practice and learn some of these additions that he struggled with. I'll do this mostly in the context of number talks, because these are easily differentiable. I can put four problems on the board that increase in difficulty, and ask students to mentally work on all four. This could give Tom a chance to practice "6+8" while giving Casie a chance to try "46+68" in her head.

**In Conclusion:**

- What here would you do differently?
- Should I be giving more individual feedback to Tom? How would you give it?
- Why else might Tom be having trouble here, besides for the ideas that I suggested?
- Missing from all this is the social aspect -- how Tom interacts with other students in class. How do you think I could represent this?
- Oftentimes I'm unconvinced by a piece of writing about teaching because I worry that it's an unfaithful representation of the classroom. Do you feel convinced that this representation of my class is faithful?

## Sunday, September 28, 2014

### 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.

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

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?)

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:**- Forgetting its feasibility, is this a worthwhile goal?
- What's your grand challenge?

## Monday, September 22, 2014

### 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.

---

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.

---

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.

## Sunday, September 21, 2014

### 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.

## Saturday, September 20, 2014

### 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:

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

Thoughts?

## Wednesday, September 17, 2014

### How I'm Trying To Mentor Myself

*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:

- Write about anything of interest for a moment or two.
- Make a list of classroom incidents that might be interesting to dig deeper into.
- Write down some questions.
- 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.
- I make a little box for any significant takeaways from the analysis.
- 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.

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