Dog Park: 100 Doors Problem
Why is this called Dog Park? What do Dog Parks and Math have to do with each other you ask? Well….
Many nights a week, Bella and I join our friend Ginger and her dogs for Happy Hour at the hidden park, where the dogs lope about unleashed, chase balls and each other, living large . . . or small, as my Bella does, while dog parents stand around, toss balls, or walk laps on the cement sidewalk encircling the grassy play space chatting and watching our pups--making sure mine does not decide to explore the two ungated exits.
One evening, all the pups and I went to the park without Ginger. Oh, how we missed her. Chatting to oneself looks a bit insane to the outside observers. I caught up with my daughter’s friend Dahlia and heard about her next steps toward college, as a rising 5th grader asked about her school sweatshirt, proudly shared he swims at her school’s pool, and added “and I love math.”
“Why do you love math?” asked Dahlia.
I was curious about his answer. I wondered if he loved math the way I do -- the puzzlement, the challenges of a good unknown, the surprises found when one digs deeply, cracking the code and applying it to other number sequences or problems, playing with patterns, finding the way to always win a game by understanding the logic and number theory behind it, or just exploring things one never even realized were math -- I leaned into his answer.
“Because I like being done first!”
I knew my ears just lied to me. He did not say that.
“I’m sorry. What did you say?” raced out and when he repeated “being done first!” my heart sank. I raised my hand to my mouth - finger over lips and thumb under the chin as if to hold my lips shut, as I often do in meetings when I am afraid my emotionally charged inner thoughts might escape my lips too soon.
My inner monologue raged loudly in my head, as my face peacefully smiled at him “Seriously? Are you kidding me?!? You did not just say that!?! Math is not about being first or being fastest! O!.. M!… G! Just NO! Have you ever heard of the Tortoise and the Hare? UH! He needs to be RE-schooled.!”
And out of my mouth, with an expression of curious hope came, “I have a math problem for you….if you’d like it?”
The week before, I devoured Math Recess by Sunil Singh and Dr. Christopher Brownell, and it was here I learned and worked through an interesting problem. To be honest, all of the problems in the book are interesting and inspiring, even for a PreK, K, and 1st-grade math teacher. My Math Recess book looks like an old loved story that was read and reread many a time, and it has been.
He was eager to show off his speed, so I shared the problem…
“There are 100 closed doors in a line. A person stands in front of each door. The person in front of Door One opens all the doors for which 1 is a factor. (I made sure he understood factors.) The person in front of the Door Two changes —opens or closes— the doors for which it is a factor. The person in front of Door Three changes all doors for which it is a factor of… and it continues like this for 100 doors. The big question — Which doors are open after the 100 people open and close the doors?
He was silent, clearly trying to find what would whip him ahead of the class. After a few moments, in barely a whisper, "This is hard.”
I gently smiled and looked him in the eyes, “And now you are doing math. Think hard. Find a question to help move you forward.”
I walked a lap around the park. As I approached him, “Do you see the way yet?” A shake of his head shouted he did not.
“What is your question?” Silence. I squatted down to meet his eyes, smiled, and suggested, "Find a question to ask, one that will help you CLARIFY what the problem is asking you to do,” and I continued onward.
Before I exited the park that evening, I offered the idea of a good problem taking more than a few split seconds to solve was worth being solved and added, “Looks like you have some thinking to do.” I waved over my shoulder and added, “ Until tomorrow.”
The next evening, as Bella and I walked into the park, the boy was there again and nodded to him and his mother. Jumping up and down, pulling on her shirt sleeve, he said to her, “There she is. That lady. (pointing to me) Go talk to her. She’ll give you THE REALLY HARD math problem."
My heart was filling up with joy for this child was still thinking about that problem. He was doing math. He was talking math with his mother. And he was sharing the challenge. YES!
As I rounded the last leg of the walkway, the other dog parents were questioning him as he tried to explain the problem to them.
“Here she is. Can you explain the problem?”
“I can, but it is your problem to solve. If you can explain it, you can solve it.”
Now picture four Dog Dads standing around a young boy of 9-10 years old as he tries explaining the math problem. They debate the different approaches with him.
I rounded the corner and asked, “Do you have a question yet?”
“100 seems big.”
“Question?”
“How can I solve this?”
“Ah, that I can but will not answer that for you” and looked directly at the group of Dog Dads standing there, “or anyone. Looking back at the boy, what joy is there in taking away your joy of discovery and learning.
“I am trying to figure out which of the 1 to 100 doors are open after each door either opens or closes if it is a factor of the starting door, like 2 is a multiple of 2 and 4 and 6, so I open or close those door?”
“You are very good at explaining the problem, which means you understand it. Now, I have a question...Have you tried using smaller numbers,” and I took a lap. My smile did not fade that evening.
As I again passed the young mathematician, I heard, “After one, all the doors are now open?
“Is that your question?”
“No.”
“Is that right?” he asked.
“Why do you think it is right?”
He turned back to the Dog Dads and had them ACT OUT Door One. Dog Dads were moving as he walked down the line of four Dog Dads, pointing at each “door” saying, “One is a factor of one, open, please. One is a factor of two. Open, please. Three is a factor of three. Open, please. One is a factor of four. Open please.”
“Ah, you are acting it out. Is it turning out to be a very useful strategy?”
“Yes!”
“You HAVE shown what happens when Door One does the work…. “(I am silent and waiting for him to suggest his next step.)…
He turns to the Dog Dads and says, “Let’s do Door 2.” I smile and move on for I was no longer needed.
On my next lap my mathematician is pushing his little sister on the swing, explaining the problem and what he has learned so far.
As I turned away from them, Daisy’s dad asked, “For the first ten doors, Doors 1, 4, and 9 are open. I think the next open door will be 16. Is that right?”
“Is that right is a good question. Why do you think you are correct?”
“The open doors 1, 4, and 9 are square numbers,” he quietly shares, using his fingers to demonstrate, “Door 1 is one squared. Door 4 is two squared. Door 9 is 3 squared, so Door 16 is four squared. And then 5 squared is door 25. Each open door is a square number.”
I smiled, “I understand your thinking. What is the answer to the original question- How many doors are open?”
I wanted to jump up and down and scream and shout. I was so excited. I admit I did a little arm pump and a “YES!”
In Dog Park, I met a random boy and shared a problem with him. Now the Dog Dads are pondering the problem, alongside the boy who has become a young mathematician, who is still working on the problem. The Dog Dads didn’t give it away, did not take away his learning when they so easily could have. YES!
Beyond being joyful over the community of dog lovers digging into a meaty problem, I am more excited than ever to find the next Dog Park challenge.