Friday, September 3, 2010

Things That May Not Look Like Math... But Are

Math is a tool we use to make sense of our world. We invented it, just like we invented the wheel or cutlery or the combustion engine.

Of course, math wasn't created out of thin air. Early math involved notches on sticks as a way of counting and keeping track of objects, possibly for trade or barter. Math then evolved into keeping track of things by 1:1 correspondence using counters to stand in for physical objects. Eventually, the objects became bundled into groups so that it was easier to keep track of large amounts. Then people started making marks in clay (as there were too many counters to drag around). In India, they developed numbers beyond 1 and thus our current number system came into being... and the rest is, as they say, history.

All this to say, there was still a lot of thinking in the human brain about how to manage tasks prior to the language of math being developed. We are not "tabula rasa" when it comes to mathematics because we are natural born problem solvers. And math was developed because people enjoy solving problems. Math is simply a tool for solving problems. And any problem has any number of possible solutions.

Young children are phenomenal problem solvers. They are fearless in their use of trial and error as they explore the limits and natural laws of the world. As they grow, we adults tend to take this innate problem solving creativity and corral it into boxes (sometimes called "curriculum") and the end result is that our kids stop seeing problems the same way. Instead of there being any number of possible solutions, they start to look for the one "right" answer.

So, what if we delay the "boxes" phase and, instead, look at all the problem-solving math-like activities our kids are already doing?

A child with a bunch of wooden beads and a piece of string will likely play with it in both predictable and creative ways. They will put the beads on the string. They may try to make it into a necklace. Or a snake. One of the first things I noticed about my son's play with beads was that his creations evolved over time. At first, the beads were placed on the string randomly. Then he'd start making simple patterns of alternating colours. Then he started to make more complex repeating patterns. Then he'd make patterns that reversed. Then he'd make patterns that alternated. Believe it or not, making patterns is actually part of the Provincial Learning outcomes and primary math teachers worth their salt will make sure that kids have this sort of activity available to them.

In addition to patterning, kids are also working with number concepts. They may be exploring 1:1 correspondence, they may be starting to group things together according to number (i.e. five beads of red) and may be learning about operations (such as multiplication, which is really just repeated addition) by repeating groups (ie. three groups of five beads is 15 beads altogether). They may be learning to skip count (which makes learning multiplication facts so much easier). There is all sorts of "math" going on without any top-down instruction or workbooks.

Blocks

The humble building block is an amazing way for children to begin to put some of their innate problem solving skills into action. When my son was younger, we had a couple of sets of Brio blocks (one set plain, one set coloured) that we spent hours and hours playing with. When he was really little, it was enough to see how high he could build a single tower of blocks before they all fell over. But it didn't take long before he was more interested in creating structural integrity - he wanted to build something that didn't fall over easily and that he could preserve for awhile. He started to to want to make his creations functional objects where he could play with toys, with doors,windows and floors. The final products were very pleasing and beautiful.

A few years ago, we found Kapla building planks, which took our block building to a whole new level. You can see the possibilities by going to the Kapla site and looking through the gallery.

The Math involved in block building has to do with patterning, ratio, geometry, spatial reasoning, fractions, measurement, and number. It takes planning and continual problem-solving to create a structure, and block play also helps kids build an intuitive understanding for concepts that will later have complicated formulas layered on top of them.

Resources: Obviously, I am a big fan of Brio blocks (as they have different shapes such as cylinders, rectangles, circles and arches), but if they are hard to find, any nice, wooden block set will do. And this is one instance where more is definitely better. Blocks are relatively inexpensive but the returns can be great so it's okay to invest in a large quantity. And they aren't just for little kids either. We still like to pull out the blocks from time to time.

You are also a resource. Kids love to build with a buddy -- an understanding adult, who won't knock a creation over on purpose, is a valued building companion.

I also highly recommend the Kapla blocks (or planks). These are much more expensive than normal blocks but worth it. If you want to jazz up your plain wooden set of Kapla, you can buy small sets of coloured planks. These small sets are not enough in themselves, though, to build anything satisfying. You will need the 280 plank set at the minimum. Even better is the 1000 plank set (comes in a lovely wooden storage box with wheels). If you buy from Mastermind Toys (in Ontario), the shipping is free for any order over \$75 (and these sets are heavy, so this is a deal).

Some people like to build with their Jenga blocks. Not only is Jenga a great game, the blocks have a nice size and weight for creative building. If this is the route you decide to go, definitely get 2 or 3 sets.

Lego

Lego bricks are blocks that stick together and have a bit more versatility due to some fancy parts. So, all of the mathematical thinking that is involved in building with blocks is also a part of building with Lego. In fact, most construction toys (such as K'Nex, Straws and Connectors, TinkerToys, Zome, Fischertechnik, Uberstix, ZOOB, etc.) require this type of thinking (so feel free to fill your home with them - more about construction toys when we get to Science).

There are a couple of things that set Lego apart. First of all, Lego has standardized units of size. There are the 1x2 bricks, 2x2 bricks, 2x4 bricks, 2x8 bricks (and so on). Right there, you can see a mathematical progression. You can also see grouping (multiplication) and ratio. Add on patterning, fractions and spatial reasoning, and you've almost got a full math curriculum, minus the pencils and the paper drudgework.

The other amazing thing about Lego is the way the instructions are created: completely pictorial with no words. A child figuring out a Lego instruction booklet is involved in some heavy duty spatial interpretation and reasoning as they take 2-D pictures and recreate them into 3-D reality. Remarkable, really.

Resources: Once upon a time, I was that parent who said, "I can't stand the contrived Lego kits where kids build toys and never take them apart... where's the creativity in that?" I've since changed my tune and here's why. The instructions, for one thing. I can see all the incredible learning that happens through the process of following them to make a completed object. The second thing is the complexity of building. By building a pre-designed object, my son has picked up on design strategies and skills that he then applies to his own creations. It hasn't detracted from his personal creativity, it's added to it. The third thing is that *most* (not all) Lego builders use the pieces from sets to make their own cool creations. And some of the special pieces you get in sets are difficult to source elsewhere. So, if your child really wants that Millennium Falcon, getting it for him or her isn't a bad thing. You can buy pre-designed sets at a local toy shop or online at Lego.com (which has the full selection).

That being said, Lego Dacta (the educational division) has fantastic sets of free-play Lego that are also wonderful to have laying around the house, especially when friends come over (if the sets are too precious to share or creations are too special to take apart). You can buy Lego Dacta sets through Spectrum.

You can also design your own Lego creation on the computer by using Lego Digital Designer (it's free!). In a perfect world, you can then order your designed set, complete with step-by-step instructions, which you then pay for. We haven't figured out how to do this yet on our Mac (as we seem to have some difficulty connecting to the store), but we will sort it out soon. Regardless of whether or not you ever buy the set, just designing one like this is a great idea.

Origami

The math in Origami is astounding (symmetry, geometry, pattern, etc.). The actual theory behind it is complex and fascinating. We recently watched a PBS show about Origami titled Between the Folds. It tells the stories of 10 artists and scientists/mathematicians who are into paperfolding. One of the mathematicians interviewed on the show is Erik Demaine, the homeschooled boy from Nova Scotia who became MIT's youngest professor at the age of 20. Dr. Demaine has a page about math and origami here (if you want to get into it). And here is an interview published in 2003 that tells Erik's story of how he got to where he is now.
 Gandalf by Origami Artist, Eric Joisel
Resources: In addition to the video (which is gorgeous to watch), there are many many books about origami at the library. You can also buy square paper at an art store (origami paper isn't necessary but it can be nice to have special paper) or just be prepared to trim regular paper to make squares. When kids first start folding, they may want to try all kinds of things. Don't be afraid to let them go "off the page" and make their own ideas. My son recently created a whale out of paper that he came up with on his own and it looks just as good as anything from a book. It's the playing with folding and unfolding that brings the learning forward, not finishing products a certain way.

And... for those kids who like a neat, finished product, we found a couple of books on origami at a museum gift shop that we quite like and are probably a fine place to start in terms of getting a handle on how the folds work.

Easy Origami by Didier Boursin.

Monster Origami by Duy Nguyen

Extra Treat: Fantastic TEDTalk with Robert Lang about Origami (and the 4 principles)

Puzzles

Puzzles are the ultimate problem-solving environment and they come in all shapes and sizes. When we do puzzles, we use an assortment of logic-based strategies, including trial and error, to come up with a solution that work. From "little kid" puzzles (like Laurie brand) to complex jigsaw puzzles to Sudoku to codes and ciphers to manipulative puzzles like Rubik's Cube to board games to video games, there is a vast array of puzzles to bring into your home and let your kids explore. Puzzling is a great past-time and it's part and parcel of mathematical thinking.

Resources: I'm reluctant to suggest puzzles with so many choices available, so I suggest going to a store that sells puzzles and check some out first hand. If you live in Victoria, Interactivity, the fabulous games store on Fort Street across from the Bay Centre, has a good selection of puzzles to peruse. Munro's Books on Government Street has a section in the adult part of the store for things like Sudoku (and related) number puzzles (as well as language-related puzzles). Look for logic puzzles, too. Usborne has a nice selection of puzzle-solving books for kids that we've quite enjoyed over the years.

By the way, I do think that every home should have a Rubik's Cube in it (and perhaps a solution book, too - it's okay to do that, really). And if you want to peruse other manipulative puzzles, here's a link to Puzzlemaster.ca, which has an incredible selection of puzzles (the Hanayama Puzzles are fantastic, by the way). When choosing a puzzle, be sure to start with an easy level if your child has not puzzled before. Puzzles can be quite frustrating at times and it's best to ease into complexity.

Board Games

Board games seem an obvious choice when it comes to math because of using dice and cards and having to add things up and subtract. In fact, psychologists have done studies to determine which games help most in developing basic numeracy skills. But there are more things mathematical to board games than meets the eye. Games like checkers, chess, Chinese checkers, and Go (as well as many other well-designed board games) require the development of contingencies and multiple strategies while holding a number of probabilities in our heads and "reading ahead" in the game. "One of the most important skills required for strong tactical play is the ability to read ahead. Reading ahead includes considering available moves to play, the possible responses to each move, and the subsequent possibilities after each of those responses." Most games also help us develop some level of frustration tolerance and encourage flexible thinking as we often need to shift our strategies throughout game play.

Although these skills may not be listed in a government learning outcome list, they are the underlying skills that we need to be effective, determined, and creative problem solvers. And that's what mathematicians are.

 Go
Resources: Suggesting specific board games seems like a daunting task, so I will list some of our favourites and some places to buy games and then let you explore on your own! Note: None of these games are special "math" games - you'll find those listed in a different post. Also, pick up a Hoyle's Rules of Games for classic card game and other game ideas. There are also dexterity games which may not seem like math games, either, but these games provide a physical understanding for some of the underlying theories and concepts in both mathematics and physics.

Classic games: checkers, chess, Chinese checkers, Go, Marble Solitaire (or Jump!), Cribbage, Ludo (Parcheesi), Backgammon, Snakes and Ladders, Mancala, Dominoes, and so on.

Quality board games for younger kids (ages 5 and up): Labrynth Jr., Hey Where's My Fish!, Zooloretto, Carcassonne (skip the expansion kits until your children are older), Elfenland

Quality board games for older kids (ages 8/9 and up): Carcassonne, Elfenland, Labrynth (or Master Labrynth), Mystery of the Abbey, Pickomino, Scotland Yard, Settlers of Catan, Thurn and Taxis, Ticket to Ride, Zooloretto

Dexterity games: Animal upon Animal, Pitch Car, Bamboleo, Bausack, Jenga

Common Games: Monopoly, Uno, Dutch Blitz, Mille Bourne, Pit

Where to buy: To find special board games in town (and to talk to a knowledgeable and friendly gamer), visit Interactivity on Fort Street. He has a good selection and he's a very helpful guy. To buy online, I like The German Board Game Company. They have a great selection, reasonable prices, a fixed shipping rate, and they are Canadian! Drexoll Games in Vancouver also has a fantastic selection of games and puzzles and the staff are helpful, but the prices are a bit higher (and you still have to pay set shipping). If you are in Vancouver, it's worth a trip to 4th Street to check out their store.

When you play a TCG, such as Magic or Pokemon, you are doing mathematical calculations all the time. They may seem like basic skills, but team that with the strategy you learn along the way while making split-second decisions about numbers, and it definitely supports your child's sense of number and operations (add, subtracting). These games provide the opportunity to develop fluency with mental calculations, which is an invaluable skill.

Video Games

I also mentioned video games under puzzles. And they are puzzles and promote logical thinking and problem solving skills. There are also games that require a good deal of mental calculations and numerical planning to play. I wax eloquent about video games here. There is also a lovely post here about the math involved in World of Warcraft.

Sports

Okay, I'm talking about watching and following sports, rather than playing them (although I suspect playing them has some math, too). Anyone who participates in a play-off pool or follows the stats of a team or certain players has to manipulate a good deal of math to do so (and has to learn to average, etc). And they don't even care that they are doing math! It's a great thing to support if your kid is into it.

Real Life

There are a number of "real life" situations that kids are immersed in (without coercion) that add to their mathematical understanding of the world:
• playing with containers in the bathtub or at the kitchen sink helps them learn about measurement and volume
• so does playing in the sandbox or at the beach
• making cookies helps them learn about measurement and fractions and estimation (how many cookies can we get from this dough?) and division (how many cookies do we each get?)
• trying on shoes helps them learn about measurement
• having an allowance and bank account (and when old enough, odd jobs) helps them learn about money and finances
• so does going grocery shopping with a parent who engages the child in conversation about choices and money and weight and volume
• rearranging their room helps them learn about measurement (as does helping to paint their room, if they are old enough to enjoy it)
• helping to build a tree house or a bird house helps them learn about measurement and fractions
• tracking Olympic medal winners helps them learn about data management (charts, graphs) as well as numeracy (calculations)
• just having lots of different types of clocks in your house and calendars (have a family calendar that kids can see) and schedules (such as flight arrivals and departures) helps them learn about time
• monitoring their own growth (height marks on the wall, the bathroom scale) helps measurement become tangible and personal
• gardening (deciding where to put rows, how far apart, how long, how many seeds per row, etc.) helps them learn measurement and, if you get into it, data management
• playing an instrument helps them understand tonal relationships, which are mathematical
• crafting, sewing and knitting require measurement, 3-D spatial reasoning, and number operations
• planning outings helps them figure out things like outside temperature, tide levels, distance, and time
• and there are likely lots more!

On Pat Farenga's website, he has a PDF booklet titled Unschooling Math that you can download for more examples about how children use math in every day life. Sandra Dodd has a page on Unschooling Math on her website as does Joyce Fetterol.

The above may seem like iffy math (with nary a worksheet in sight!), but math came from our desire to manage and understand our world... to solve problems. The algorithms (adding, subtracting, etc.) and formulas we think of as being "mathematics" are laid on top of our problem solving abilities. The more problem solving experience we have that is meaningful and real, the more meaningful (and memorable) mathematical concepts and constructs will be to us.