Learning isn’t Linear

This is Bea. On top of the slide.

Bea made it to the top of the slide by walking up the slide itself, the “wrong” direction, as most children like to go. Another detail to know about Bea is that she just turned one-year-old. Oh, and- she’s not walking on her own, yet. Well, at least across horizontal surfaces. Okay, she made it up the slide by using her hands to grip while she walked her legs up (not exactly a full bipedal move), but I have to ask how many of us would think to give Bea a crack at walking UP the incline of a slide when she hadn’t even demonstrated confidence and proficiency with moving upright across the flat plane of the living room floor? But Bea was ready for a chance at the slide, and the only assist she needed was removal of her socks. I appreciate every time I am reminded that learning isn’t linear. We can’t map out a perfect progression of learning that every child will follow step-by-step. We need to be willing to let the learner have the lead and follow where they take us, even if–no, especially if–they are headed somewhere we didn’t think they could go. This post might not seem like it is about math, but I promise you it is.

Big Eyes and the Element of Surprise

Harry Potter Mystery Mini-figure toyTo market anything to the 10 and under crowd these days all you need to do is add some big eyes and the element of surprise. For my son (10) and my niece (8) this holiday Harry Potter Mystery Mini-figures became the obsession. With one figure each in hand we started our seven store sold-out toy search for more. I eavesdropped in on their backseat chatter as we traveled from store to store.

Harry Potter Mini-Mystery Figure Box with probability for each figureThe back of the box kept them occupied as they began to categorize the figures into four groups: 1) common – 1/6 odds 2) uncommon- 1/12 odds 3) rare – 1/24 odds and 4) epic – 1/36 odds. They went on to discuss their ideas about how many you would have to buy to get all 12 figures in this series. They also had boxes from 2 different series, which both included Harry Potter, of course. “How many Harry’s would you get if you tried to collect all the figures?” they wondered. And, “What are the chances we both get an ‘epic’?” and “What are the chances we get the same figure?” Somehow, nobody but me was wondering, “How many stores will we have to visit?” or “How many miles have we covered so far?” Occasionally they turned to the calculator on my phone to work out some of their reasoning. I have no idea if their math was correct. This is partly because my attention was on the road, but also because it didn’t really matter to me in that moment if it was. This was a mathematical conversation between two kids who were motivated to explore this situation. What more can I ask? I work hard to be patient and accepting of math ideas, trusting that correct methods will show up with time. I don’t need to do anything in this moment, but listen. Even by asking a question I might run the risk of derailing the conversation, or zapping their energy.

Thankful for my Barnes and Noble educator discount and holiday gift money from family, they each left with two more mini-figures. After working out the rules for how they would open them (each of them went to a separate room and called out the category of probability before revealing) on the way home, they finally opened their boxes. The dreaded duplicate…a painful life lesson in experimental probability. And yet, another opportunity to work out rules about trading and fairness of trades based on the commonality of the figure. Odds are there are more mystery mini-figure math discussions to come in my house.

Noticing Math Noticings

Do you remember your child’s first word? Or maybe you’ve heard about the first words you used early on in life?

My parents tell me the first sentence I uttered from my high-chair was “Hi, man! Or was that a woman?” as a neighbor with flowing 80’s metal hair passed by the kitchen window. I know other things about my early literate life- my favorite book was The Little Engine that Could, I liked to listen to Little Toot on the record player, and my favorite story to hear at bedtime was The Lion and the Mouse (yes, I see a theme). But, I can’t tell you what math ideas I was tinkering around with at the same age. It’s not something anyone noticed, or remembered to share with me later.

I don’t think my parents are that different from most grown-ups. We are really good at noticing and acknowledging new words and language kids are taking on as they grow. But do we pay attention to early math learning in the same way?

By the time I had my own son I was working in education as a math specialist, but without a background in early learning I felt clueless about these earliest years. I certainly didn’t have time to read-up on early math as a busy working mom. So I just started noticing- paying attention to moments I thought might be math-y, even if I couldn’t exactly describe why, yet.

These moments happened in lots of situations I didn’t expect to be mathematical. One night we were reading before bed. We read lots of Sandra Boynton (I can still recite most on demand) and Blue Hat, Green Hat was up next.

Blue Hat, Green Hat book cover

Mom: “Which do you like better, letters or numbers?”

Kid: “Letters. Green hat has one more.”

I think he counted, maybe he visually compared. In any case my son was quantifying the letters on the cover of his book- something I wouldn’t have thought to do. Bedtime is book time, after all, time for reading and telling stories. And if I was asked to pick out the math ideas in this book I would have thought about attributes (color) or position, but not quantity. Children notice so much more than we do; by listening to their ideas we can often see something new.  

I realized in that moment that one of the simplest things I could do is just start noticing what my son was noticing. It was late, well past bedtime, and I just wanted to get finished with this “one last book.” It wasn’t the right time to talk about his math idea, but by noticing I was starting to gain a sense of this little mathematician living in my home.

So I set out to notice. Inside, outside, playtime, lunchtime- I started asking myself, “What is the math in this situation? What is my son paying attention to now?” Thinking about every situation as novel and unique became mentally challenging very quickly. So I shifted gears and started thinking about simple math categories I could look for in any situation.

Quantity, Size, Position, Shape Imaage

  • Quantity– Is he paying attention to how many, or how much?
  • Position– As he plays is he going under, in, on, or behind?
  • Shape- Does he ask for the one that is round, or pointy?
  • Size– Is he talking about ‘tiny’ or ‘huge,’ or ‘too small’ or ‘too tall’? Does he notice ‘shorter’ or ‘longer’?

I think about how quantity, position, shape, and size are being used during play, or to communicate. I see him comparing, sorting, or making patterns based on these qualities. With just a few categories to think about, it became easy to notice everyday mathematical moments while we were doing just about anything.

When he is older, I will be able to talk with my son about his favorite books, the bedtime stories he begged to hear most. I will also be able to share what kinds of math-y things he was up to through the years.

16 Cents a Day

When my son asks a math related question like, “How much longer until we get there?” or “How many baseball cards are in the whole box?” I do the good teacher-mom thing and come right back at him with the same question: “I don’t know, how could we figure it out?” Despite his proclivity for math, the predictable groan, eye-roll, “just tell me!” sequence inevitably follows. These are often problems I know he could solve quickly, with little effort (like, why is he even asking in the first place), and I know he genuinely wants the answer. Yet, he resists doing the math.

And then there are moments like this summer morning when my son is idly lounging on the couch watching TV:

Dad: “I hear they have a lot of golf gear at Costco. But, we aren’t members any more, right?”

Mom: “Right, but it’s only $60 for a membership. It might be worth it to join again.”

Dad: “That’s alright. I can shop somewhere else.”

Voice from the couch: “It’s only 16 cents a day, Dad.”

Mom: [pauses to do the math] “Uh, yeah, just 16 cents a day. Seems reasonable.”

I resist the urge (and it’s a strong one) to ask how he decided it was 16 cents a day. Did he think about 60 x 100, and then divide by 365? Did he divide $60 by 12 first? I don’t probe, I just notice. And move on.

I wouldn’t have considered prompting my son to solve this problem. In fact, I had barely noticed there was a math problem involved. The choice to take-up this problem belonged to him, so there was no resistance. While I don’t imagine I will stop sharing problems I find relevant or interesting, nor will I start readily providing answers for questions I know my son can answer himself, I am trying to be mindful of following his lead. Letting kids choose their own moments to math seems important. Especially in the summer.