Now Online: Becoming the Math Teacher You Wish You’d Had

becoming-the-math-teacher-you-wish-youd-hadTracy invites you on a journey through this most magnificent book of stories and portraits…This book turns on its head the common misconception of mathematics as a black-and-white discipline and of being good at math as entailing ease, speed, and correctness. You will find it full of color, possibility, puzzles, and delight…let yourself be drawn in.

— Elham Kazemi from the foreword

While mathematicians describe mathematics as playful, beautiful, creative, and captivating, many students describe math class as boring, stressful, useless, and humiliating. In Becoming the Math Teacher You Wish You’d Had, Tracy Zager helps teachers close this gap by making math class more like mathematics.

Tracy spent years observing a diverse set of classrooms in which all students had access to meaningful mathematics. She partnered with teachers who helped students internalize the habits of mind of mathematicians as they grappled with age-appropriate content. From these scores of observations, Tracy selected and analyzed the most revealing, fruitful, thought-provoking examples of teaching and learning to share with you in this book.

Through these vivid stories, you’ll gain insight into effective instructional decision making. You’ll engage with big concepts and pick up plenty of practical details about how to implement new teaching strategies.

All teachers can move toward increasingly authentic, delightful, robust mathematics teaching and learning for themselves and their students. This important book helps us develop instructional techniques that will make the math classes we teach so much better than the math classes we took.

Add comment December 15th, 2016

Comparisons: A Little Bit More Older

In this fun guest post from Tracy Zager, you can follow along as her daughter tries to figure out just how much older some of her friends are, and as she does, you can get an excellent insight into how Tracy guides her mathematical thinking. Tracy’s new book Becoming The Math Teacher You Wish You’d Had will be published in 2016.

Ask any elementary school teacher, and she or he will tell you that comparison problems are much harder for most kids than operations with other actions. For example, fourth-grade-teacher Jennifer Clerkin Muhammad asked her students to draw a picture of this problem from Investigations:

Darlene picked 7 apples. Juan picked 4 times as many apples. How many apples did Juan pick?

Her students are pros at representations and skillful multipliers, but we saw a lot of this:

apples unsure

Kids who were used to thinking of multiplication as either groups, arrays, or repeated addition had some productive, furrowed-brow time ahead of them. How can you represent comparisons? How is this multiplication? How do you draw the 4?

Then they started to get somewhere.

apples groups

apples array

With each representation they discussed, Jen asked the excellent question:

“Where do we see the 4 times as many in this representation?”

Seriously. Jot that one down. It’s a keeper.

Jen and her class played with a series of problems involving multiplicative comparison, (height, quantity, distance, etc.). For each problem, she asked student to create a representation. And then she’d choose a few to discuss, asking, “Where is the ______ times as ______ in this representation?” Lightbulbs were going off everywhere as students broadened and deepened their understanding of multiplication.

Teachers often see the same struggle with comparison situations that can be thought of as subtraction or missing addend problems, like:

Marta has 4 stuffed animals. Kenny has 9 stuffed animals. How many more stuffed animals does Kenny have than Marta?

If kids have been taught “subtract means take away” or “minus means count backwards,” then this problem doesn’t fit the mold. And if kids have been taught keyword strategies, they’re likely to hear “more,” pluck those numbers, and add them together.

Knowing this as I do, I jump on opportunities to explore comparison problems with my own kids. I want them to have a fuller sense of the operations from the beginning, rather than developing too-narrow definitions that have to be broadened later. So, subtract doesn’t mean take away in our house. They’re not synonyms. I actually don’t worry about defining the operations at all: we just play around with lots of different types of problems, and I trust that my kids can make sense of them.

Yesterday, I was talking with Daphne about a neighbor visiting tomorrow. Our neighbor has a little girl, whom I’ll call Katrina. She’s about 2. We also have more awesome girls who live across the street. Let’s call them Lily, 15, and Gloria, 13. Gloria helps out once a week, and my girls idolize her. Here’s how the conversation kicked off:

Me: “If Katrina’s mother comes over to help me on Sunday, could you play with Katrina for a bit so her mom and I can work?”

Daphne: “Yes!!! We love playing with her. And she loves us. She cries when she has to leave us. We’re her best friends.”

Me: “I think Katrina feels the way about you and Maya that you and Maya feel about Lily and Gloria. It’s fun to hang out with older kids. You learn a little about what’s coming for you.”

Daphne: “Yeah. But for Katrina, Lily and Gloria would be TOO old. Like, she’d think they were grown-ups because they’re SO much older than she is. We’re older than Katrina, but we’re less older than Lily and Gloria are.”

So there’s my opportunity. She was comparing already. With a little nudge, I could encourage her to quantify and mathematize this situation. I thought I’d start with the simplest numbers so she could think about meaning first.

Me: “How much older are you than Katrina?”

Daphne: “I think she’s 2. So…”

There was a pretty good pause.

Daphne: “Are we counting on here?”

That’s paydirt.

Me: “What do you think?”

Daphne: “I think so. Because I want to know how many MORE. So, 2, 3, 4, 5, 6. I’m 4 years older than she is. Yeah. That makes sense.”

Me: “OK. So how many years older is Maya than Katrina?”

Daphne: “6.”

Me: “You didn’t count on that time. What did you do?”

Daphne: “Well, I knew that I was 4 years older than Katrina, and Maya is 2 years older than me, so 4, 5, 6. Because Maya is OLDER than me, I knew they had to be farther.”

Me: “What do you mean farther?”

Daphne: “Like, it had to be more. I was starting with 4, and then it had to be more farther apart.”

Me: “Oh! So you were trying to figure out whether to go 2 more or 2 fewer?”

Daphne: “Yeah. And it had to be 2 more, because Maya’s more older than Katrina than I am. I mean, I’m older than Katrina, and Maya’s older than me, so she’s more older than Katrina than I am. It had to be 2 MORE.”

Hot damn!

Daphne’s life might have been easier right here if she had some tools she doesn’t have yet. I’m thinking about number lines, symbols, and a shared mathematical vocabulary including words like distance or difference. Those tools might help her keep track and facilitate communicating her thinking to somebody else. But I love that we were able to have this conversation without those tools, while driving in the car where I couldn’t even see her gestures. Her thinking is there, and clear. The tools will come as a relief later on, when the problems are more complicated.

We kept going.

Me: “OK. So you’re 4 years older than Katrina and Maya’s 6 years older than Katrina. I wonder how that compares to Gloria? Is she about the same amount older than you two as you two are older than Katrina?”

Daphne: “Well, she’s 13. This is gonna be hard.”

Long pause.

Daphne: “She’s 7 years older than me.”

Me: “How’d you figure that out?”

Daphne: “I went 6. 7, 8, 9, 10, 11, 12, 13.” She held up her fingers to show me.

Me: “So how much older is Gloria than Maya?”

Daphne: “Umm…so Maya’s 8…umm…5. She’s 5 years older than Maya.”

Me: “How do you know?”

Daphne: “Because Maya’s 2 years older than me, so it’s 2 closer.”

This is the kind of problem where kids lose track of their thinking easily. Do we add or subtract that 2? In the last problem, she’d added. This time she subtracted. And she knew why!

When I glanced at Daphne in the rear view mirror, she had this awesome look on her face. She was looking out the window, but not seeing what was outside because she was so deep in thought. I wish I could have peeked inside her mind, and seen what she was seeing. Because, to use Jen’s question, where do we see the 2 years older? How did she see it? There was no time to ask, because she was onto pulling it all together.

Daphne: “So, Gloria’s a little bit more older than us than we are older than Katrina, because Gloria is 5 and 7 years older, and we’re 4 and 6 years older. But it’s pretty close.”

We pulled into the school parking lot on that one.

In our family, we count and manipulate objects all the time. Counting more abstract units like years, though, gives me a chance to open the kids’ thinking a little bit. In this conversation, Daphne was both counting and comparing years. Tough stuff, but oh so good.

(This post first appeared on Tracy Zager’s blog.)

Add comment August 11th, 2015

What if you weren’t afraid of math?

The majority of elementary school teachers had negative experiences as math students, and many continue to dislike or avoid mathematics as adults. In her inspiring speech at ShadowCon during this year’s NCTM conference in Boston, Tracy Zager asked the audience to look at how we can better understand and support our colleagues, so they can reframe their personal relationships with math and teach better than they were taught.

Tracy is the author of the upcoming book Becoming the Math Teacher You Wish You’d Had: Ideas and Strategies from Vibrant Classrooms (to be published in 2016 by Stenhouse Publishers). She also works with schools as a coach/consultant, and loves learning together with teachers and students over time. Watch a video of her speech below and then read her calls to action and see what you can implement in your school/classroom.

Call to Action

  • This is a collection of words mathematicians’ use when they talk about mathematics. Discuss it with your colleagues, making an extra effort to include everyone.
  • Each person should choose a word that appeals most to him or her. It should be a word that’s not currently a big part of your math teaching and learning, but you wish it were.
  • Using your colleagues as resources and collaborators, make the word you chose central to the planning, teaching, and learning of your next lesson. Don’t skimp on the conversation with your team–that’s part of the point!
  • Teach the lesson. Afterwards, get back together with your colleagues and talk about it. What was different for each of you? What was different for your students?
  • If it works for you, consider sharing the image and exploring it with your students as well. There are lots of possibilities here.
  • Write a few paragraphs here so we can learn together. Describe what happened and what you learned. What ideas do you and your colleagues have for building on this exercise?

Add comment June 9th, 2015

A Brief Ode to Blank Paper

Prescriptive worksheets can often cheat children out of real thinking and understanding. Read how Tracy Zager encourages her own children to explore and communicate their mathematical thinking through writing—on simple blank paper. And then hop over to her blog to read how she continually makes math a  natural, fun, and important  part of her family’s conversation.  

A Brief Ode to Blank Paper

Maya (7) brought home a folder full of completed math worksheets yesterday, which put me in a funk. First, there was the bugs problem.

IMAG3939-1I couldn’t decide which part of this problem bothered me most. Was it the ridiculous premise? I mean, come on. Was it that the problem was, yet again, a multiple-choice question? Was it the stubborn insistence on drawing bugs with the wrong number of legs? Was it that students have no room to work, but the publisher took plenty of room for cutesie drawings? Harumph.

As awful as the bugs problem was, the page that really upset me was this one:

IMAG3936Sigh. A few issues:

  • The strategies are named wrong, and made to look more complicated than they are.
  • There are better and other ways to solve this problem.
  • Students are left with 2″ to do math on this entire page, which is barely enough room to do the standard algorithm. There’s no chance they’ll try out one of the other strategies with no space to work.
  • Once again, we’re turning useful, general computation strategies into prescriptive algorithms. Breaking the numbers up by place value and adding the partial sums becomes: Step 1) Add hundreds…

I’m not the first math teacher to notice that prescriptive worksheets are a problem. Kamii, the CGI group, and others have all written about it. Yesterday, though, I was upset as a parent because Maya has stellar, flexible mental math skills, and her instinct to think is being undermined by this curriculum. I asked her why she’d opted for the standard algorithm on all the problems, and she said, “Those other strategies are too confusing.” I covered up her solution to 597 + 122 and asked her to solve it mentally.

“Well, I’d give 3 to the 597 to make it 600. Then 600 + 122 is 722. I’d take the 3 back, so it’s 719.”

I did the same thing for 209 + 376:

“200 + 300 is 500. 500 + 70 is 570. 9 + 6 is 15. 570 + 15 is 585.”

I pointed to the top of the page and said, “You just used this strategy. You broke the numbers up into place value parts, and then added each part together, starting with the biggest part.”

Her jaw dropped.

And my mind clicked. She has made NO connection between the mental math strategies she uses with fluency and all this junk on the worksheets. The reason? She’s never been given the chance to record her own thinking at school.

I think I’ve decided what one of my bigger problems with this curriculum is: they never use blank paper. They never write 209 + 376 at the top of a big piece of paper and let kids have at it. The kids never get a chance to wrestle with keeping track of their thinking or figure out organizational strategies. All math problems are either on worksheets or educational technology. The kids just don’t write enough.

So now I know what to do with Maya at home. We’re going to spend some time with blank paper, where she has to work out how to write down what she does in her head. She needs to make mistakes, lose track, not be able to follow her own thinking, and then ultimately figure out ways that make sense. She needs to be able to write down her thinking so that she and her mathematical community can follow it. I’m on it.

Two hours later, after dinner, Daphne (5) got us started. Our dining room chairs have decorative nailheads, and the kids are forever running their fingers over them and counting them. Daphne said, “Someday, I’m going to get out a math journal and count all these nailheads and write it down so I know how many there are.” Before I knew it, she was off! Someday turned out to be right then. Both kids got in the game.

IMAG3946-1

Daphne was incredibly excited to count AND write down her results. Check them out:

IMAG3950-1If you want to understand her notation, take a peek at the short videos:

The power of blank paper, baby.

Perhaps inspired by all of this discussion and my venting, Maya asked if she could get a piece of blank paper when she did her homework, which is truly a counter-cultural act with this curriculum. “Of course!” I nearly sang.

IMAG3960-1-1

She created this number line and used it to solve the final problem.

IMAG3951When I asked her about it, I pointed out that she didn’t just answer the question by saying, “The red one.” She wrote about the problem more generally: “I used a number line and found that anything less than 350 would fit and 270 is less so red paint is less than 350!”

She said, “Well, when I wrote it myself I thought about it more.”

Precisely my point.

 

This post first appeared on Tracy’s  blog.

3 comments March 5th, 2015

“You just listened, so then I could figure it out”

We recently had a lovely post from Sarah Cooper about the importance of taking things slow in the classroom. Today’s post from Tracy Zager is based on the same idea — giving students the space and time to figure things out on their own. Tracy was reminded of this important lesson by her daughter during a car ride. Read her story and watch the video and then head over to visit Tracy’s new blog!

“You just listened, so then I could figure it out”

My daughters and I climbed in the car to go shoe shopping before their first day of school. I sat in the driver’s seat while they buckled themselves into their car seats and noticed I was keeping track of the loud clicks I heard for each buckle. I took the opportunity to open a math conversation with my kids.

“I didn’t look, but I know you’re all buckled. How could I know that?”

Daphne, age 5, said, “You looked in the mirror!”

“I did not look in the mirror!”

Maya, age 7, said, “You must have counted the clicks! So, you heard 6 clicks and knew we were all buckled.”

I asked, “How would 6 clicks tell me you’re both buckled?”

Maya answered, “Because each car seat has 3 buckles, and 3 times 2 is 6.”

Daphne started to cry. “That’s what I was going to say!”

I turned to Daphne. “Tell me where the 6 comes from, in your own words.”

Daphne said, “Each car seat has 3 buckles, and 3 plus 3 is 6.”

Ah! There was my first opening.

“Maya, you said 3 times 2 is 6. And Daphne, you said 3 plus 3 is 6. Can those both be true? They sound different.”

Maya and I played with this idea for a few minutes, but I could see in the mirror that we were losing Daphne. When Maya and I were done, I asked a question just for Daphne.

“Daphne, what if we had 3 car seats? How many clicks would I hear then?”

There was a long pause while she thought, Maya waited, and I drove.

“Nine!”

“How did you figure that out?”

“Well, I remembered the 6, and then I said 7, 8, 9.”

Maya gasped. “Daphne, you’re counting on again!”

Daphne beamed and said, “I know!”

We were all excited because Daphne had counted on for the very first time that morning, when we were baking popovers.

I asked, laughing, “Since when are you counting on? How did you learn that?”

Daphne said, “Well, you gave me a lot of time to think. You didn’t say anything, and you didn’t tell me what to do. You just listened, so then I could figure it out for myself.”

My jaw dropped. For the rest of the car ride, Daphne talked about how school should be filled with lots of time when the teacher “doesn’t say anything and lets the kids think, because that’s how we can learn. The teacher can just listen.” There was so much wisdom in what she was saying that I asked her if we could make a quick video once I parked the car.

 

Daphne knows what she needs to learn math: time and something tricky to figure out.

“Do you like when a problem is tricky?”

She nodded.

“How come?”

“Because then I get some time to think, and I learn something.”

I am a teacher, and I also coach other teachers. How many times have we all talked about think time, and how important it is? But, here’s the truth: about halfway through the time Daphne was thinking about the fourth car seat, I got a little nervous. I tried to keep my face encouraging on the outside, but on the inside I heard a tiny voice:

“Uh-oh. Maybe this problem is too hard. Should I help her? What would be a good question to help her?”

While I was secretly worrying, Daphne was calmly figuring out how many clicks 4 car seats would make. To a teacher who makes decisions every few seconds, 20 seconds of think time—which is what Daphne took to solve this problem—feels like an eternity. New teachers, in particular, tend to break silences after a second or two with some kind of “help.” With practice, I’ve learned how valuable think time is, and I now sustain those long silences. But internally, I still find it hard to quiet that worried voice.

Later that night, after we watched the video together, I asked Daphne about the reason she gave for why counting on is challenging. She’d said, “You have to remember while talking about something else.”

“What did you mean by that, Daph?”

“Well, you have to remember a whole bunch of things. Like, I had to remember the 6, because that’s where I started. And I had to remember the 3, because I had to stop after 3. And I was counting at the same time. It’s a lot to remember!”

“It sure is. Can I tell you something? While you were doing all that, I was wondering if I should help you.”

She looked shocked. “But I didn’t need help, Mommy! I was just thinking!”

“What would have happened if I had said something while you were remembering where to start and where to stop while you were counting?”

“I would have forgotten what I was doing and had to start all over again! That wouldn’t have helped at all, Mommy! That would have been so frustrating!”

“You know, Daphne, you’re making me a better teacher. You’re teaching me, again, that sometimes when teachers want to ‘help’ a student, we’re actually not helping at all. Sometimes we just need to be quiet. And we need to be comfortable with silence.”

“Yeah. So kids can think!”

“Yeah.”

We were quiet for a minute together, each thinking.
“Mommy, can you tell other teachers that too? Tell them what I taught you? To not interrupt us when we’re thinking, and just listen while we figure it out?”

“Yes, honey, I think I can.”

Tracy Johnston Zager is the author of the upcoming Stenhouse book Becoming the Math Teacher You Wish You’d Had: Ideas and Strategies from Vibrant Classrooms. You can find her on twitter at @tracyzager, and at http://tjzager.wordpress.com.

Add comment September 17th, 2014


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