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.
Svetlana Pancenko was inspired to start an after school math circle in London called Igramatica when she realized that all children are mathematicians, and can have the joy of exploring mathematics through games. There is math all around us, and Svetlana helps young children discover it. Here is Maria Droujkova’s interview about this adventure.
Can you describe a meeting of your group? How do you plan, what do the kids and adults do, and what happens after?
My colleague and I usually discuss and plan each lesson: the structure of the lesson, what games we’ll play and in what order we’ll play them, and what materials and handouts we need to prepare. During the lesson we always alternate active games with board games so that the children do not get bored or tired. The introductory part of each lesson is a fairy tale where all of the characters solve different mathematical and logical problems. We chose this way of presenting new topics because children like fairy tales and acting. You can keep the kids focused and entertained while including them in the process of problem solving along with the characters.
The main things we do in our math circle are different active games, board games, construction, and combination. For our construction games we use lolly sticks, bricks, LEGO, mosaics, tangram, ornaments – either with instructions to follow or with using the imagination, which children like more. Symmetry has been our favorite last year. I can see how kids get the rhythm and enjoy the beauty of what they build. The combination part is about finding different combinations: how many handshakes are between 5 friends, how many roads are between fairy tales character’s houses, how to make a lion-goose or rhino-lion, how many ways we can build a pyramid out of three different colored rings.
Each game aims at developing the children’s mathematical, logical, and analytical thinking using their imaginations and abstract thinking. That’s why these lessons are suitable for kids with different abilities.
All parents are welcome to participate in the lessons and play with their kids in the group. We believe most parents need to learn how to play with kids so that they can play at home, on vacations, or during long trips.
After the circle the kids have some time to play whatever games they want with each other. For example, they can use a whiteboard for drawing, or build anything they want with colored sticks. Meanwhile we discuss with parents (privately or in group) what to focus on mathematically with their kids at home. We discuss what the kids achieved, what their strengths and weaknesses are, and where they need a little extra help.
What made you want to start a children’s program? What helped you to start doing it, once you decided?
We were inspired to start a math circle by reading Zvonkin’s book “Math from Three to Seven.” At first our circle was only for two boys. It was obvious to us that the best way for kids to learn is through play and competition in games. So we examined methods of mathematical teaching in different countries: The UK, Russia, USA, China, and Japan. We made comparative research and learned about the most interesting experiences. We tried many different games with our own kids. We saw how enthusiastic they were during these lessons, how they enjoyed themselves, and most importantly, we saw how easy it was for them to understand fundamental mathematical concepts through playing games. During the lessons we could observe kids’ reactions to each game and understand which games they liked, which needed to be corrected, and if we needed to make the games more challenging or more simple.
Then our group gradually expanded: we invited our kids’ friends, neighbors, and classmates. All the kids were excited to come and play with math, listen to new stories, and participate in competitions or team games. So we decided to organize an after school club where kids could play and learn math.
Can you describe a memorable episode or anecdote from your meetings? Something that warms your heart! (You can share more than one such story.)
Some days I feel really energetic and satisfied after the lesson when everything I planned worked and the kids happily played and asked for more.
At one of our first math circles about fractions we cut up small pancakes and then ate all the pieces. When I decided to go back and refresh fractions for the kids a couple months later the first question from one boy when he heard the word “fraction” was: “Hurrah! Will we cut and eat pancakes again?”
Another story: we told a fairy tale about a little ant who hurried back home and counted the legs of bugs and spiders. In our math circle we made the bugs and spiders out of green and red grapes and toothpicks. Another “tasty story”, and the kids learned which creatures have 6 legs and which have 8.
In one lesson at the beginning of our circle I introduced coordinates to 4-5 year olds. It didn’t go to well until we used a chess board with small lids from water bottles and hid tic-tacs under them. The kids were intrigued to find and eat the tic-tacs, and this helped them understand where H2 or A8 was!
What other inspirations can you name?
We are happy to be a part of the Moebius Noodles network; it gives us so many new and fresh ideas on how to develop our club further. Our other inspirations are: Jane Katc’s Moscow based club which holds a lot of family math camps in different countries (http://janemouse.livejournal.com/), board games of “Banda umnikov”( http://bandaumnikov.ru/), and games shared by Nataliigromaster(http://nataliigromaster.blogspot.com/).
If you could go back in time, what advice would you give yourself for working in groups?
The most difficult question :) I would advise myself to make my lessons more structured and use connections between them, and to not be afraid of repeating things and making them a bit more difficult each time.