Moving Through Math: two book reviews
Learning math through the arts… Moby Snoodles approves! So I was happy to review two books from Marcia Daft’s Moving Through Math project. They are read-aloud books for kids under five and their grown-ups. Both books are illustrated in a distinctive, memorable manner that reminded me of early Japanese watercolors. Of math education aspects, I found notes to parents to be most distinctive and memorable. After a strong book, you can graduate from “What?” (what entities and actions were in the book) – to “So what?” (your new understanding of the world).
Count, Drum and Shake It
What? Use iconic symbols, such as clapping hands or shaking maracas, to form pattern units, such as “clap, clap, shake.” Then repeat the unit to form the pattern. Perform each movement as you get to that symbol in the sequence.
So what? Unitizing is the foundation of multiplicative and proportional reasoning. For example, 5*3 can mean a unit of five repeated three times, or a unit of three repeated five times. Kids need to work and play with units a lot, such as repeating parts of songs and dances. A kid who thinks of any unit as a single building block is well on the way to the idea of variables.
Thumbs up!
- The book uses iconic symbols – that is, pictures that show what to do. Iconic symbols help kids transition toward abstract symbols. Iconic symbols are accessible even to babies. If the baby is too young to clap or jump, you can move the baby’s hands in yours, or bounce the baby in your lap as you point out the symbols.
- The relatively advanced idea of pattern units is introduced qualitatively (without numbers) first. This helps learners to form strong, grounded foundation for the idea. Qualitative intros work for learners of any age, but are absolutely necessary for young kids.
- “Look for patterns in the air, look for patterns on the ground.” For example, walking feet: left, right, left, right… Or flying wings: up, down, up, down… Inviting kids on scavenger hunts after introducing a new math topic? Yes!!! Scavenger hunts help to notice many aspects and details of a topic; they make the topic stick in memory; and they integrate math with the everyday world of learners.
Building on it
- Do more multiplication. In particular, invite kids to multiply within pattern units. For example, how do you double the pattern unit “clap, clap, shake”? That is, how do you show 3×2 in the language of the book? “Clap, clap, shake; clap, clap, shake” is what the book does. You can also do “clap, clap, clap, clap, shake, shake”!
- Make the idea of variables explicit. Give your pattern units names. For young kids, the traditional X and Y are too short and abstract for variable names. Invite kids to create and name several pattern units. Kids often name things after themselves, their favorite heroes, or their pets. Let’s say we have these names for pattern units:
Alice = “clap, clap, shake”
Bob = “drum, clap”
Then kids can make patterns out of Alice and Bob!
Bob, Bob, Alice = “drum, clap, drum, clap, clap, clap, shake” - Provide even more support so kids can make their own patterns by transformations of old pattern units. The book suggests two transformations. Reflection turns “clap, clap, shake” into “shake, clap, clap.” Swap between moves turns “clap, clap, shake” into “shake, shake, clap.” Invite kids to make up their own transformations. Or catch their mistakes in repeating patterns, and turn mistakes into transformations!
Meadow Count
What? A traditional counting book, for numbers from one to ten. Children count by making whole-body movements, such as three giant puddle jumps, seven high reaches to the clouds, or eight jiggles down as you wiggle to the ground.
So what? Fine motor skills use some of the same parts of the brain as mathematics. That’s why activities that require fine motor skills, such as playing musical instruments, crocheting and origami, develop “the math brain.” But you don’t want to overwhelm and overload this math brain. Challenging fine motor skills, on top of challenging new math ideas – that can be too much for one activity! That’s why using gross motor movements helps to introduce new math ideas. Whenever I work with kids, I design whole-body methods for introducing each topic, from equations to infinity. This way, the math brain can devote itself to math, without having to coordinate the fine motor movements too.
Thumbs up!
- Children in the book look and dress like they come from all different continents, without any cliches. Kudos to the artists on this tasteful presentation!
- One-to-one correspondence is not challenging as an idea: babies are born understanding it. However, the reliable implementation is hard for young kids. Gross motor movements will definitely help to count more reliably.
- The descriptions of movements are artistic. There are evocative metaphors, such as “count one leafy spin,” and emotions, such as “two happy hops,” and details that go with the story, such as “nine sharp tugs to create a bouquet” when the kid picks wildflowers. Kids love that rich stuff – and relate to math through it!
Building on it
- Use more intrinsic quantities to support the number sense. One example in the book is to hop two times to take off both shoes. You could also sprint to each corner of a room (four sprints) or reach to the sky with each finger on your hand (five stretches).
- Connect counting to pattern units. For example, count to nine again and again as you create many bouquets for all your friends.
Reviewed by Maria Droujkova
Posted in Grow
1+1=2 but Mostly it Doesn’t
Have you had a chance to look through the 1+1=5 and Other Unlikely Additions by David LaRochelle? If you (or your child) think that addition is booooring, make sure to get this book. Once you do, get all nice and cozy, but leave plenty of room for your child (and possibly yourself) to jump up and down from excitement. That’s how great this book is. Oh, and by the way, forget about learning how to add two numbers. That’s really not what this book is all about.
It’s a little bit about arithmetic, but mostly it is about iconic quantities and set theory. If you simply must have some counting thrown in for good measure, there’s that too.
A while ago we shared our Iconic Numbers Multiplication game. We also briefly mentioned that playing with iconic numbers helps children develop number sense. If you are interested in the how and why of this, check out this iconic arithmetic primer.
In that same post we also suggested a variation of the game in which you look for examples of iconic addition instead of multiplication. We also mentioned that iconic addition is harder to find than iconic multiplication. Except it’s not if you read 1+1=5! In fact, once we were done reading this book (and, at my son’s request, re-reading it several times), we started seeing iconic additions everywhere!
1+1=7 since both the dining room light fixture (6 bulbs) and the living room table lamp (1 bulb) were on while we read it
1+1=1 since I held the book with one right hand (0 band-aids) and one left hand (1 band-aid)
1+1=3 since Luke Skywalker fights with 1 light saber while General Grievous fights with 2 (ok, this doesn’t occur in the actual Star Wars story, but happens regularly in our family room)
This activity is very addictive and might lead to speaking in puzzles that are a blend of 1+1=5 and “I Spy”:
I spy with my little eye
One plus one now make a five
Ask that around the dinner table and see who’s the first to guess it right (just make sure to place a fork and a knife next to each plate).
Next thing you know, you’re raking your brain trying to come up with rhymes for other numbers. I quickly gave up on rhyming and switched to a simple “I see a one plus one that makes a ten. Do you see what I see?”
The book inspires kids (and adults) to see everyday objects as sets, or collections of other objects. For example, a triangle can be viewed as a set of 3 sides while a rectangle is a set of 4 sides. An octopus is an example of a set of 8 (arms) while a starfish hides a set of 5 (arms) in plain sight. If one set has 8 elements and another set has 5 elements, then when added, the two sets have 13 elements total. Hooray!
So yes, there’s some addition in the book. But again, it’s not central. So don’t worry if your child is not yet ready to add 100 and 10. The illustrations are wonderful and help even very young children enjoy the book.
Posted in Grow
Last-Minute Christmas Decorations and Tangled Lights
Days are flying by and everyone is busy preparing for the holidays. While you might not know what gift(s) you will be getting this year or whether the kids will catch the sniffles right before the party, one thing you can count on is tangled lights. Seriously, no matter how carefully we pack them, when we open the boxes next year, here they are, all knotted up. Turns out, it’s all about math and physics as opposed to our carelessness. Check out this NPR article for details. Here’s the best part – next time you have knotted up strings of lights, give them to your little one to undo and proudly tell your friends he was busy solving a knot theory problem.
In addition to tangled lights, another thing you can count on is a broken ornament or two or three. If your tree looks a bit bare, don’t rush to the store. Instead, consider some really cool last-minute DIY ornaments. Bonus is they are all about math. The easiest ones are paper chain garlands. This year, add a twist, (ok, half a twist) and turn them into Moebius chain garlands. Or bend pipe cleaners into shapes and grow Borax crystals on them. What shapes you choose is totally up to you. Inspired by Vi Hart’s Borromean Onion Rings video, we made this Borromean Rings ornament.
Or how about turning cardboard boxes you were about to throw away into star ornaments just like Malke and her daughter did on the Map is Not the Territory blog.
Finally, we’d like to share a holiday printable we found thanks to the link from one of the readers, Elena T. She printed it for her daughter to color days ago, but we just got around to it. This Christmas tree might seem like just a giant cute coloring page, but don’t let it fool you. It’s got lots of things going on, including some great math. Can you find examples of gradients, tessellations, pattern, rotational symmetry, radial symmetry, iconic quantities and a lovely Sierpinski triangle?
Share your holiday math with us on the blog and on our Facebook page.
Posted in Make
The Round Table Quest
A few days ago I decided to introduce my son to the first of the Sir Cumference books. The entire series is one of the most popular living math books and came highly recommended by quite a few of our math friends. So I sat my son on the couch and told him we were going to read the book about the Knights of the Round Table, something he previously heard about only in a Magic Treehouse story.
I am not sure what I wanted out of it, but a thought of my little one asking me to re-read the story before running off to his desk and feverishly drawing and cutting out shapes did cross my mind. So I was a bit disappointed when he did not ask and did not run off to try it all out with the compass and the triangle. Oh well, maybe he’s not yet ready, I thought.
Except later that day, as my son was playing with his toys on the kitchen table, he asked me “Mom, wouldn’t it be nice if we had a round table in our kitchen?” And that’s how it all started. For a while we talked about the benefits of a round table over our rectangular table. After all, we are a family of three, so we have plenty of room around our smallish table.
My son argued that when grandparents visit, there isn’t much room left and two people have to share one side. And he wanted to see if cutting our table into a circle would make things better.
First we needed to figure out how to find the center of our table which was a nice little geometry problem. It might had been too difficult for him to solve on his own. But I reminded him that he already knows how to find a center of a rectangle because that’s what we do when making origami jumping frogs.
Since we couldn’t fold the table, we had to simulate the folds with some yarn and tape. We then marked the center with a push-pin.
Next we had to figure out how to draw the largest circle we could. The terms we came across in the Sir Cumference story – diameter and radius – came in quite handy in this discussion. My son tried using our compass, but quickly realized it wasn’t big enough. And we didn’t have any round object big enough to be traced. So I showed him how to make a compass with a pushpin, yarn and a pencil. The first few circles we drew were not really circles since we were yet to learn the lesson of keeping the yarn stretched throughout the project.
In retrospect, this was the perfect time to stop and compare the lengths of yarn in taut and slack states.
Once we got a more or less acceptable circle drawn on the table, we needed to make its circumference more visible. I didn’t want to damage the surface any further, so I asked not to use markers or crayons. The only other option was blue painter’s tape. So we brought a roll of tape out, unrolled a long piece and tried marking circumference with it. Except it didn’t quite work.
Which meant we had to figure out how to solve the problem of marking something rounded with something straight. I didn’t think my son would figure it out, but almost immediately he suggested to tear off little pieces of tape instead of one long one. And so we did. We tested different lengths to find a good balance between accuracy and speed. Still, that was a lot of pieces of tape and after a while my son delegated the work to me and simply observed the process. But it was also a great opportunity to recall the Greedy Triangle story.
And then we looked at the finished project and compared the perimeter and area of our would-be round table to our existing rectangular one. It was still big enough for three people, but just about. Trying to fit grandparents was out of the question. So we took the tape off (turns out, it does leave marks on wood, so beware) and decided to leave our table as is.
Posted in Grow
