I recently had the chance to play around with the newest book here at Natural Math called Avoid Hard Work…and other Encouraging Mathematical Problem-Solving Tips for the Young, the Very Young, and the Young at Heart, by Maria Droujkova, James Tanton, and Yelena McMannan.
I’ve been around Natural Math for coming up on two years now. I think I own every book we’ve published! I am not entirely done with the book but I believe this is one of the best books Natural Math has put out.
Because it isn’t another collection of math activities that I have to figure out for myself how to adapt. This book walks you through every aspect of the activity-how to do it, why to do it, how to adapt it, why to adapt it that way, how to improvise, how to see math in it for me, and more. My first reaction was “Finally a book that helps ME learn how to ‘do’ Natural Math at home!”
There are so many wonderful books and sites with collections of activities or motivational books about math, but every time they have left me just a bit disappointed that I as reader wasn’t quite sure how to make use of them. Avoid Hard Work feels like an owners manual or a chef’s training. It’s where I learn principles in a specific context but I can easily use what I learn in a new contexts. I hope those who read it will feel much more confident in their ability to lead Natural Math adventures. I believe you will!
My four, almost five year old, and I tried out the activities in Wishful Thinking (chapter 3). (Spoiler Alert – skip the first photo if you don’t want to see the solution!)
This is his solution to the initial problem. When I gave it to him, I explained the rules, but I was really curious to see what HE would want to do with it. He asked if the lines could “jump over each other.” I asked him what he thought. He decided that wouldn’t really work on paper only with string. He asked if we could go “behind” the bubbles on the top. I told him we could, but let’s see what we could do if we didn’t first. He was up for that, so we started out by drawing lines and erasing them. He was excitedly chatting about how it wouldn’t work if he drew this line or that. He even erased a huge line that he decided wouldn’t work. I was happy when he simply said, “Oops! That way wasn’t it. Let’s erase it and try a different way!” (All that growth mindset and mazes work has paid off!) I guided him in finding this solution, but after drawing B, he did C and even helped me finish A.
Here’s what happened next! He decided he wanted to try it again with different letters. Letters that were more meaningful to him! Letters from our names and letters he currently finds interesting. This time he revamped the rules to allow for going behind the bubbles on the edge of the paper.
If you rearrange the order of the bubbles and they are all on the edge it totally changes the challenge! I loved letting him make his own rules up. : ) Next he tried it again and got more creative with his lines.
Probably the best part for me was just spending some time with him. He felt so accomplished and happy with this activity. He loved telling me what he was doing and why he was doing it. He even challenged his older siblings ages 14, 12, 11, 10 and 8 to try it out at dinner. He loved knowing the solution before they did. It was cute as they had to ask him not to give hints so they could try it on their own. The older ones enjoyed it too! Some got a solution faster than the others which caused mild upset (always a challenge with many children), but eventually they all found the solution and asked for more problems like it!
No matter the ages of your children, I’d encourage you to order your copy of the book today! It’s an invaluable guide to making Natural Math a part of your life, home or classroom.
Lydia Gordon is a math circle leader who used our Inspired by Calculus materials in several sessions, and sent these fun notes. Here are a couple of math sparks that go with activities mentioned here, in case you want to try them too: How You Slice It and Make Shapes with Sticks.
Have some math notes to share? Write us!
Here are Lydia’s notes.
When we discussed dimensions, these were questions that worked well with our explorations of playdough, sticks, fruit slices, and other building materials:
About derivative as taking-apart: “Not the first thing on my to-do list – to get blown up. It’s, like, my nine hundred thousandth!”
“She made some minor modifications. And by minor, I mean a lot.”
A boy named Quin, integrating a bunch of banana slices into a 3D banana: “It’s genetically modified through Quin-gineering!”
Math circles need warm-up activities in several situations. One is the beginning, when everybody still arrives – an insight puzzle. It is solved by playing with dimensions, the point of view children can develop from calculus activities. We started with a brain teaser from The Moscow Puzzles: 3 toothpicks are connected with playdough to form am equilateral triangle. Can you form 7 such triangles with 9 toothpicks? Everyone started arranging toothpicks in 2D patterns. In five minutes, my seven-year-old built a pyramid (3D) which sparked the idea of building in 3D and led children to the eventual answer.
Another warm-up was for brainstorming our construction activities. We talked about different ways of creating something. Out-of-the-box examples came up: Quentin Tarantino creating Curious George in his style, a tree trimmer performing brain surgery, or The Muppets singing Smells Like Teen Spirit.
Pretend-play with toys is how we invite children to consider new mathematical points of view. The kids built circles with different numbers of Keva Planks: 8, 16, 32, and 100 (diameter was ~5’). We walked a LEGO mini-figure along the different circles, as if they were slices of the Earth. Children saw why the Earth looks flat to a small person walking it. We noted that the wedge of space between the planks became smaller as the circle became larger. We ended up building with Keva planks the rest of the time. The challenge I gave them was to build a cantilevered bridge. Two teams built each side of the bridge and met in the middle.
The kids and I enjoyed having a math problem at the beginning of the meeting. I will definitely be using The Moscow Puzzles again. I’m about halfway through Mobius Noodles, and that was a great resource too.