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

Posted on by tzager

This blog was also posted at Stenhouse’s blog.

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 during 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 four 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 six, because that’s where I started. And I had to remember the three, because I had to stop after three. 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.”

I’m not impressed.

Posted on by tzager

(An earlier version of this post contained a picture of a child’s work, the child of a friend. I have checked with my friend, and she has asked me to remove the picture, which I am happy to do. I should have asked her permission from the start. I am sorry I didn’t. I have also removed my discussion of the picture, and written a different beginning to this post. I’ve kept the original comments, though.)

I wear two hats. I’m a teacher, and I’m a parent. I see the world of children and learning from under both brims. One pattern I notice is that everybody gets excited by student learning, but we often get excited by different aspects, and use different language to describe what we see.

Among my teacher friends, we talk about moments where we see a child trying to make sense, so we can analyze it. What made it possible? What’s the child thinking? How can we understand what he or she did or said? Where’d that thinking come from? What would be a good question to ask next? For example, this year on twitter, I shared this picture and my daughter’s thinking about it:

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When I asked her how many meatballs there were, she counted the top row, 4, and the bottom row, 4, and said 4 + 4 = 8. Then she counted the left column, 3, and the right column, 3, and said 3 + 3 = 6. She skipped over the middle 2 meatballs, added 8 + 6, and told me there were 14 meatballs in all. When I asked her about her count, she said,

“I counted it that way because, when you count arrays, you count the corners twice.”

She had me count the array so she could explain. When I counted 4 meatballs in the top row and 3 rows down the left side, she said, “See, you counted that corner one twice!” I had an aha! moment because I realized she picked up that misconception when watching me count arrays. Her mistake helped me understand her thinking, but also my teaching. I shared it on twitter, and we had a lovely chat about it.

Among fellow parents–on facebook, at the playground, and at birthday parties–I hear very different talk about children’s learning. I’m generalizing here, of course, but I hear a lot more talk about accomplishments than I do about thinking; a lot more talk about achievements than I do about mistakes. I hear a lot of fixed-mindset language, much of it designed to impress other parents:

“He’s very intelligent. I mean, he’s doing double-digit multiplication already! I don’t remember doing that until 3rd or 4th grade!”

“She’s such an avid reader. She read Harry Potter when she was six!”

The desired response, and the one I tend to see and hear, is some variation of, “Wow! That’s so impressive!” Used this way, “impressive” doesn’t mean, “that’s really thought provoking and memorable, and has made an impression on me.” It’s more like, “Wow, your child is advanced/smart/ahead!”

So what does that usage say about learning, and our goals, and what we value?

If the goal of students’ learning is to memorize and recall facts, trivia, procedures, data, vocabulary, then we have plenty of markers and milestones, a whole ruler full of age-based hashmarks. Parents can be impressed when their children say or write or do (I can’t bring myself to use the word “learn” here) something that seems “advanced” for their age. “Jenny knows all her capitals already? Impressive.”

If the goal of students’ learning is for them to understand ideas and concepts deeply, and to build connections among those ideas, then we care less about those tangible achievements that impress friends on facebook. From the parents’ point of view, we have learning where the adult gets to listen, join in, interact, enjoy, or participate, rather than evaluate and compare to an age-based yardstick.

For example. One day on the way to kindergarten, my then 5-year old daughter Maya said, “4 and 2 make 6, and 3 and 3 make 6, and 5 and 1 make 6. How will I know if I’ve found them all?”

This is an amazing question. Was I impressed? Hmm. That’s honestly not the word I’d use. I was excited! I knew her question was the beginning of something good, not the end. I am usually more excited by questions than answers, and, “How will I know if I’ve found them all?” is such a mathematical question to ask! We spent much of the rest of the car ride exploring different wrinkles of it. For example, once she had found them all, she decided to figure out how many combinations there were:

“Mommy, is 4 and 2 the same or different as 2 and 4?”

“What do you mean?”

“Well, I am counting how many ways to make 6, and I want to know if they’re the same thing, or if I should count them separately.”

“What do you think?”

“Well, sometimes yes and sometimes no.”

“What do you mean?”

“Well, I’ve noticed that, when I add, it doesn’t matter which number I start with. 4 and 2 is 6, and 2 and 4 is 6. I usually start with the bigger number because then I have less to add on so it’s easier, but it doesn’t matter because I end up with 6 either way. It doesn’t matter what order they’re in.”

“Then why’d you say sometimes yes and sometimes no?”

“Because if I have 4 stickers and Daphne has 2 stickers, it’s different than if I have 2 stickers and Daphne has 4 stickers! We still have 6 stickers altogether, but it’s different because one of us gets more and one of us gets less, depending on which one has 4.”

There are so many rich and wonderful mathematical ideas in this conversation: the commutative property of addition, contextualizing and decontextualizing problems, counting strategies, permutations and combinations, and so on. Yet, the numbers weren’t “impressive” or “advanced,” were they? We were talking about sums to 6, numbers I think Maya chose because she could compute them comfortably in her head.

I tell all the parents I know about the Talking Math With Kids blog because Christopher does a lovely job showing adults how to engage in a conversation like this with children. Tell less. Ask more. Listen. Value the child’s thinking.

I think the first step for parents might be to evaluate less and join more.

If the outcome of a conversation about your child’s thinking is either to feel impressed or disappointed, then you are evaluating.

If the outcome of a conversation about your child’s thinking is that you and your child now know each other a little better, you understand how your child is making sense of the world, and/or either or both of you sees an idea in a new way, then it sounds like you joined.

If the conversation was short, and involved the child showing or telling you something and you remarking on their learning, you evaluated.

If the conversation was longer, rambled some, could have drifted off or kept going or even resumed after a break, then you probably joined.

If you left the conversation eager to tell someone else what your child did or said because you think that person will be impressed, you evaluated.

If you left the conversation thinking and wondering, then you joined.

If we were evaluated by our parents as kids, we tend to evaluate. Joining takes practice. It might not feel natural at first. But I am not exaggerating when I say these incidental conversations about learning–the ones that happen while my kids and I are walking down the street or driving in the car or setting the table–are some of my most treasured memories as a parent. The best way I know to invite these conversations is to stop evaluating how our kids compare or measure up, and just listen to them.

Updated:

I loved this quote, in a comment from Ed:

From the memoir of Nobel Prize winning physicist, Surely You’re Joking, Mr. Feynman:

He … taught me: “See that bird? Its a Spencer’s warbler.” (I knew he didn’t know the real name.) “Well, in Italian it’s Chutto Lapittitda. In Portuguese, it is Bom Da Peida. … You can know the name of the bird in all the languages of the world, but when you’re finished, you’ll know nothing whatsoever about the world. You’ll know about the humans in different places, and what they call the bird. So let’s look at the bird and see what it is doing — that’s what counts.” I learned very early from my father the difference between knowing the name of something and knowing something.

Elementary Teachers as Math Learners

Posted on by tzager

One of the central messages of my upcoming book is that elementary teachers have an incredibly hard cycle to break. Almost all of us were taught math very badly. We memorized procedures. We learned rules. We circled keywords. We did it the teacher’s way. Now, we’re all grown up and are expected to teach math for understanding. Yippee!

Except we don’t understand.

In my coaching and writing, I spend a lot of time creating safe spaces for elementary teachers to engage with the powerful and fascinating math we teach, so we can build deep, connected, conceptual understanding. It’s my life’s work to help teachers learn together and alone, from each other and with our students.

But what happens when it’s me? What happens when I don’t understand?

This question came up for me on Twitter today. Kristin asked:

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I looked at her question, and thought, “Of means multiply. Why? Why does ‘of’ mean multiply in fractions?” I learned that rule as well as anybody else, but, like Kristin’s students, I am struggling to make the connection. As a fourth-grade teacher, I’ve never had to teach this content, so haven’t yet dug into it enough to re-learn the math meaningfully. Here we were on Twitter, for all the world to see, and she was asking for advice from me, as if I had expertise. I thought for a minute, and then put my money where my mouth is.

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That was a hard tweet to write, for a few seconds. Then I saw Kristin’s reply:

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And we were off! So began one of my favorite Twitter conversations to date, in which several incredible teachers chimed in, especially Brian Stockus and Zak Champagne. Each of us freely admitted the concepts that were hard for us. We made sense together. We laughed. I had paper next to me, and was sketching and working out ideas and models as we talked. I emerged from today’s conversation a whole lot wiser about multiplying and dividing fractions. Feel free to read the storified version. It’s really good.

When I coach teachers about learning content more deeply, I always talk about how much bravery it takes to admit we don’t understand. Today, it felt great to be brave myself.

Updates:

Brian blogged about “of” after this conversation.

Kristin blogged about our online professional learning community.

And the twitter conversation continues: https://twitter.com/MathMinds/status/509795670530994176 This a meaty topic. We’re not done!

The Structure of Popovers

Posted on by tzager

We spent a lovely week on Mount Desert Island this summer, playing in Acadia and eating lots of popovers, which are apparently kind of a thing in that part of Maine. When we returned home, the kids asked if we could make some popovers ourselves. Happily, I dug out an old favorite, this Maida Heatter recipe, which says it makes ten popovers. We use a standard twelve-muffin tin, so I asked my five-year old daughter, Daphne, to butter all the way around the outside and leave the middle ones empty. When she was done, I asked, “How many popovers are we making?”

IMAG3021Daphne looked at the tin for a minute, and then started counting from the left-hand side, one column at a time. She looked at the first column and said “Three” without counting, which means she is now able to visually recognize, or “subitize,” three objects as a group. Then she moved her finger to the second column, counted the top cup, skipped the middle one, and counted the bottom one. “Four, five.” She kept going: “Six, seven. Eight, nine, ten.”

I found her count really interesting because I expected her to count all the way around the outside, following her butter tracks. Instead, she looked at the layout of the muffin pan, and used the grid. This is an exciting development! She is looking for some structure, rather than plunging right into counting without a plan. I’ve watched this thinking of hers grow over this summer, starting when a friend asked her, “Do you think you could count all those boats?”

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Daphne took one look and answered, “If they were in a line I could.”

Daphne is feeling a need for organization, and for developing strategies to keep track. As numbers get bigger and problems get more complex, she is looking under the numbers for structures like lines, columns, patterns, and groups. These structures are essential for the important mathematics to come, like place value and operations.

Counting doesn’t get the respect it deserves. When I watch kids count lots of things, I see them develop that need for organization. After a child tries to count a pile of action figures, sneezes on figure 31, and has to go back to the beginning because she’s lost track, she starts to think about strategies that might work out better next time. Before long, students are naturally counting out groups of 10, 20, 50, or 100, like the students in these videos: Counting Collections with Kindergarteners and Counting Collections with Third Graders. (This Teaching Children Mathematics article by Julie Schwerdtfeger and Angela Chan has a nice description of the value of counting collections in the classroom.)

In our home, we count all the time. We count anything and everything, from popover tins to ponytail holders to toys on the floor the kids haven’t picked up. After all that real and varied counting, Daphne is starting to count strategically. Once we made the batter, I said, “It looks like we have a lot. Maybe we’ll be able to fill the whole pan. How many popovers would we make if we filled these empty ones also?” I pointed to the two empty cups in the center of the muffin tin.

She thought for a moment, then sub-vocalized, “Ten. Eleven, twelve.” She turned to me and said, “Twelve!”

I asked, “Where did the ten come from?”

“When I counted it before.”

“You didn’t need to start back at the beginning and count them all?”

“No, Mommy. I remembered the ten. I know that’s ten. I just counted from there. Ten…then eleven, twelve.”

Her older sister, Maya, came barreling into the kitchen and said, “Daphne! You’re counting on! I heard you!” She explained the term to Daphne, who was grinning broadly.

I asked, “How did you learn to count on, Daphne? That’s the first time I’ve ever heard you do it!”

She said, “Well, I just counted a lot of times, and then I started remembering the first number. When I counted it again, it hadn’t changed.”

When I coach teachers, I sometimes hear primary teachers instruct kids to count on instead of count all. “I don’t want to hear anybody go back to one. Keep track of where you were and count on.” I hear intermediate teachers say, “I don’t want to see anybody using repeated addition. You need to multiply.” Or I’ll hear, “I don’t want to see any dots. Make gridlines. No more dots.”

I think about this kind of talk a lot. On the one hand, I’m glad when teachers have a sense of the thinking that’s on the horizon for the student, and it’s sometimes appropriate to nudge students toward more efficient techniques. On the other hand, I’ve never seen this kind of nudge work. What if, instead of disparaging kids’ strategies and telling students they have to abandon their trusted techniques, we think about creating the need for more organization, better keeping track, more efficient strategies? For example, if kids insist they can multiply by repeated addition, what happens if we give them some bigger numbers to multiply, and then keep quiet? The children will eventually become frustrated–exasperated even–with the laborious calculations. Now there’s an opening! Now they feel the need for an improved strategy. Now they are looking for a different, more appropriate structure than the one they were overtaxing. And now they own all kinds of good math thinking, which we get to honor as theirs. If we’ve created a classroom climate where students are respected as mathematical thinkers, they’ll be perfectly comfortable saying, “This problem was a total nightmare and I’m sure I made some mistakes doing it this way! I think I need to figure out a new way to do it.”

Daphne believed there were ten buttered muffin cups because she’d counted them herself. She believes in her new strategy of counting on because she’s verified her counts enough times to trust them. She owns her thinking. I see no skipping over these phases. If students enter school without this kind of experience, we need to provide opportunities for them to build this foundation in our classrooms. Otherwise, the student who is counting on because his teacher taught him a procedure, (whether he is convinced it works or not), is the same student who finds an answer of 6,080 instead of 680, and doesn’t bat an eyelash.

The recipe, by the way, made 12 perfectly imperfect eggy popovers, which we ate with butter and jam. Enjoy!

Welcome!

Posted on by tzager

I’m incredibly late to the blog party, but have finally shown up. Quick thanks are in order to:

The MathTwitterBlogoSphere, for being. And for helping newbies get going!

Christopher Danielson, for his blog Talking Math with Your Kids, which has opened a wonderful conversation I am happy to join. Find us chatting on twitter with the hashtag #tmwyk.

I’m glad to have this designated space to discuss ideas. I hope you’ll chime in and push my thinking!

Tracy

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