(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:
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.
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.