A few years ago I viewed “The Hundred Languages of Children,” a travelling exhibit about the Reggio Emilia approach to early childhood education. If you’re not familiar with this approach it, among other things, considers the environment (of the classroom and other spaces) as a “third teacher”.
Of course, I was drawn to the part of the exhibit that focused on movement and dance as one of the “hundred languages” with which children express themselves. There was a video that showed the children’s first experiences with an old factory space – a huge room empty except for two rows of large, white columns. The children were running around and between the columns, peeking around them, and interacting with their friends, all movements and ideas that eventually turned into a formal piece of choreography.
At the time I was just starting to think about creating a math/dance program for preschoolers and my biggest question was how could I encourage that kind of exploration? It seemed unlikely I would be able to find an empty factory or other interesting environment and get a bunch of preschoolers there on a weekly basis. And then it hit me – I could create an environment out of tape. I could define three-dimensional space using two-dimensional lines and colors.
After experimenting with my daughter’s preschool class, I came up with some starting points for parents and teachers who are interested in employing tape in the interest of math and kinesthetic exploration of space.
A simple straight line taped down a hallway becomes a pathway. It also divides the space in two, and provides a chance to walk on it or jump over it. Best of all, one can march (or walk, or skip, or slide, etc.) rhythmically down it singing “As I was marching down the street, down the street, down the street…” Or, tape two or more parallel lines down a space and see what happens when you sing “Down by the banks of the hankey pankey, when the bullfrogs jump from bank to bankey…”
A simple alteration of a child’s environment can deepen their experience and exploration of the space around them. When my daughter was three her teachers put down a straight line of tape to help the class ‘line up’ before leaving the classroom. It was a simple, visual learning strategy that appeared to work as envisioned by the teachers. Later in the year though, I saw pictures of what else the kids had done with the line. They had used their large blocks to build a wall the length of the tape and then lined up their animals and cars alongside it.
A simple taped perimeter can highlight empty space, as in “Find an empty spot inside the tape and make a shape.” Floor tape can define and redefine the space it’s in. Large open spaces encourage a lot of endless running. The minute you create a large rectangular box on the floor, with corners, you now have enough visual cues to focus a preschooler’s attention to IN (the box), OUT (of the box), AROUND (the sides of the box), CORNERS, and ACROSS, all age-appropriate math terminology.
Ultimately, I would love if every parent or preschool teacher would put down taped lines in their living and learning spaces then stand back to observe how the children interact with them.
Start with one straight line and go from there but don’t bring attention to it. Let your kids find it and interact with it on their own volition and let us know what you observe!
p.s. FYI, when I talk about ‘floor tape’ I am referring to two different products, both of them sticky. First, there’s painters tape which is blue and low tack so it can come up easily off both hard surfaces and carpet. There is also the floor tape that P.E. teachers use, which comes in lots of fabulous colors, the better to design with, my dear.
This week’s Math Goggles challenge is a perfect excuse to go out and buy a dozen of freshly baked bagels. At least it was for me. (If you are not sure what is a Math Goggles challenge, read about it here). So the question is what do you get when you slice a bagel.
The answer depends on how you slice it. That’s one reason you need several bagels. Begin by cutting the first bagel horizontally. What will the cross section look like when you finish slicing? Easy, right?
Next bagel! Instead of making a horizontal cut, slice this one in half vertically. Can you predict what the cross section will look like this time? Still easy-peasy! (And now you know the correct answer next time you see this little puzzle floating around on Facebook)
Time for bagel #3, inspired by James Tanton. Can you slice it in such a way (different from the first two) that the cross section will show two circles? Hint: choose your best-looking (most torus-like) bagel for this one.
The forth bagel can be cut into a trefoil knot which is super easy to do, but you sacrifice the ability to toast your bagel before putting cream cheese on, or rather in, it.
Four down, eight more to go. Time for the big one, George Hart’s interlocking bagel halves puzzle. I made a mistake of putting half my bagels into the freezer, so by the time I attempted this problem, I only had 2 bagels left. And I didn’t want to draw on any of them with a Sharpie. Let’s just say that I learned a few valuable lessons in the process and ended up with a failed, but perfectly edible experiment.
So, sharpen your knife, get out a tub of cream cheese, and keep your eyes open for math!
Finding math patterns or cute meanings in dates is a popular pastime. We celebrate Pi Day on March 14th, because 3.14 are the first three digits of that number. Star Wars day is on May 4th, from the pun on the Jedi blessing, “May the Force be with you!” It’s also neat to have dates like 12/12/12.
David Coffey started a meme on making every day mathematically special. Try it with your kids! Ask them for their favorite numbers, and celebrate when those days come up.
You can use a sign generator, or simply write on paper and put that sign up in your house. I made a few signs for upcoming dates, using some of my favorite math tidbits: composite numbers, Fibonacci sequence and plain old multiplication.
This week’s Math Goggles challenge is very simple – watch one of your child’s favorite cartoons and look for math in it. If it seems interesting, intriguing, strange, weird, and worth investigating, look further into it. Yep, that’s it! Here’s how it worked out for me.
My child’s most favorite cartoon of the moment is called Lego: The Adventures of Clutch Powers. I like it, but not enough to sit through all 88 minutes of it every other day. Still, I overhear snippets of dialogue as I go about doing my own things. One of the snippets that caught my attention was a trick question from the Troll Warrior to Clutch Powers:
Troll Warrior: Using nothing but standard eight-stud bricks, how many bricks would it take to build a spiral staircase as tall as three minifigs?
Clutch Powers: There are more than 915 million ways to combine standard LEGO bricks. But they will never connect on the diagonal.
Really? I mean, I understand the impossibility of connecting on the diagonal (had to explain it to my son a few times before), but 915 million combinations?! This sounds pretty amazing.
Turns out, the story is even more amazing than it sounds. The 915 million figure, or more precisely, 915 103 765, is the number of unique ways to combine just six two-by-four studded LEGO bricks of the same color. If this sounds unbelievable, check out the explanation of the LEGO Counting Problem. It is a great read and has just one formula that’s pretty easy to understand.
One of the best parts of the paper is in the “Why is this interesting?” section:
Such challenges are always important to drive mathematical research, and it often happens that methods developed to study a problem with no practical applications (like this one) are useful to study problems which do have an impact on everyday life.
Back to the original question of building a spiral staircase out of LEGOs… This is when the second part of my challenge started. And let me tell you, I totally cheated by looking at… my 6-year old’s recent builds.
He recently built a 3D treasure map using some unit blocks and LEGO bricks, complete with the “X marks the spot” signs which he made out of LEGOs, of course. That’s how I knew that, under some conditions, regular LEGO studs and plates can intersect at angles other than 90 degrees. So I used this idea and built a very basic spiral staircase in no time although it does not follow the ”use nothing but two-by-four studs” rule.
Now it is your turn to find math in your child’s favorite cartoon!
Welcome to adventurous math for the playground crowd! I am Moby Snoodles, and I love to hear from you at firstname.lastname@example.org
This week, we had the thrill of holding the full file of the book in front of our eyes. The front and back cover, Library of Congress and ISBN numbers, Table of Contents, everything! It was so tempting to rush the book to you right then! But we had to follow the plan for another round of quality assurance, with multiple people checking and re-checking everything. More than thirty corrections of layout, text-illustration correspondences, and typos later, the book is going to the printers and the ePub formatting. We will have to check on the printer’s proofs, as well. Like Zeno’s arrow, “Moebius Noodles” is approaching the error-free state. No book ever gets there, but we’ll get close enough for all practical purposes!
Meanwhile, here are three of many stages of our back cover design: the finished look with 3D models of the real opened book; the first schematic of the layout; and an early draft. The back cover is a diagram of how to navigate through the book. We thought it would be a bit more useful than the advance praise you traditionally find on back covers, which we love too, by the way. We will be aggregating reviews online.
Blogs and networks
Sol Lederman interviewed Maria Droujkova for his Inspired by Math series on notable modern mathematicians and mathematics educators. Why is there so much computer science in the “Moebius Noodles” book? What is math literacy? What is Maria’s secret plan for taking over the world? Listen to the podcast and find out!
Our Spanish-speaking readers will enjoy math videos by our book illustrator, Ever Salazar. But math is a universal language! Check out what Mary O’Keeffe (Albany Math Circle) is doing, after having found the videos via the Moebius Noodles Facebook page:
Thanks to Maria Droujkova for bringing this wondrous, brilliant, and engaging series of math videos in Spanish to my attention! It is a terrific way for me to pick up at least a little bit of mathematical Spanish before my trip to lead activities at a math festival in Mayaguez, Puerto Rico this weekend.
Math Goggles issue #7 is inspired by Keith Devlin’s MOOC (Massively Open Online Course) on mathematical thinking. It is about the ambiguity of our language, with examples from newspapers, such as “Teen found after ski slope disappearance” (where did the slope go?).
Math Goggles #8 invites you to make puzzles using LEGO and other 3D tools.
Find a math and logic puzzle that you’ve not seen or solved before. Now, build it with whatever it is you have handy – cardboard, wrapping paper and glue; modeling clay; marshmallows and toothpicks; building blocks. You might like the challenge of recreating a pen-and-paper puzzle with 3-dimensional objects. Or you might like the idea of taking a 3D puzzle and drawing it.
You are welcome to share the contents of this newsletter online or in print. You can also remix and tweak anything here as you wish, as long as you share your creations on the same terms. Please credit MoebiusNoodles.com
More formally, we distribute all Moebius Noodles content under the Creative Commons Attribution-NonCommercial-ShareAlike license: CC BY-NC-SA
Talk to you again on March 30th!
Moby Snoodles, aka Dr. Maria Droujkova
What objects can you add, and what do you get as a result? I will re-post this game once in a while, as we find more good examples. Check out what Moebius Noodles readers submitted:
- 1+1=7 One square plus one pentagon equals to seven diagonals (JJ Rodríguez)
- 1+1=1 One car traveling east plus one car traveling west equals to one crash! (David)
- 1+1=45 One mother plus one daughter equals forty-five years old (Pei Lee)
Here is an example that makes me go “awww” – found via Arithmetic Village:
Moby Snoodles says: “Add your own example!”
These are examples we have so far. It takes about five minutes for your answer to appear here. Wait and then reload the page to see.