Maria at The Homeschool Math Blog pointed out this post: Number Bonds = Better Understanding, by Denise in IL at A Home for Homeschoolers. [Update: Denise's blog has moved; that post is now here.]
This is the way we teach math too! Kids find it easier to see relationships between numbers when you show that a whole can be separated into two or more parts--and you can demonstrate that with a group of things (like a pile of blocks, as Denise's post describes), or a whole thing split into pieces (there we might get into fractions). The post and comments refer to Singapore Math, but we do much the same thing with Miquon Math and Cuisenaire Rods (picture here). [OK, that's changed too. Try this one instead--scroll down a bit to see the photo.] If an orange rod is understood to be ten units long (assuming a white rod is the basic unit), then you can make a "train" just as long as the orange rod by grouping two yellow rods (each 5 cm long), or a light green plus a black (3, 7), or a white plus a blue (1, 9) and so on. You reverse that by showing a white plus a blue and asking what rod will be just as long as that "train"--that's addition. Then you get even more complicated by showing the orange rod with a white rod underneath it, and ask for the missing piece that completes the "train." That's the beginning of subtraction. (Update: If you still can't get the idea of this, you can play with some virtual Cuisenaire Rods here--click on "Start the Integer Bars Applet." This site (www.archytech.org) calls the rods Integer Bars, and the only real disadvantage to them is that they can slide on top of each other in a way that real 3-D rods can't; so you might accidentally overlap your two yellow rods and think that they were equal in length to a blue rod instead of an orange one. The real thing is definitely better, but these at least can give you the sense of how the rods work.) [2012 UPDATE: the virtual rods are long gone--sorry!]
When Ponytails was in first grade, we got a lot of math ideas from the Miquon First Grade Diary (not a diary you write in--it is a day-by-day description of a first grade math class back in the '60's). Lore Rasmussen (the author) described some hollow wooden tubes she had, each 10 cm long, into which you could insert Cuisenaire rods--any number of them that would fit into the tube. The idea was to play a guessing game. If I put a white and a blue rod into the tube, and cover up the blue end so that you can see only the white end--and I tell you that there are only two rods in the tube--then you can guess that the other rod must be blue. The game can get more complicated when three rods are used--if you can see the colours of the two ends, what is the hidden rod in the middle?
And where do you buy these tubes? I have no idea, but I made my own out of 3 x 5 inch file cards. I cut several cards to a 10 cm length (how do you like that for mixing measurements?), folded each one several times around an orange rod, and taped them shut. Instant rod tubes. When they wear out, you can make new ones in a couple of minutes.
We played other games too that develop the idea of whole/parts, particularly relating to the number 10. One game The Apprentice played in kindergarten came (I think) from Peggy Kaye's book Games for Learning. You need a bowl or other container, and five paperclips (for a young child). The bowl goes a short distance away, and you take turns chucking the five paperclips into the bowl. When you're done with your five, you say something like, "Oh, I got one on the floor and four in the bowl. I get four points." Then the child throws her five, and you say, "Oops, you got just two in the bowl and three on the floor." The point of the game is not to get to be a great paperclip chucker, but to get acquainted with all the "parts" of the number five.
Another whole/parts activity I've done with all the Squirrelings is to take several crayons, or blocks, or any small objects, hide them behind my back, and bring some of them back out--how many are still behind my back? Then they have a turn to hide the objects. Sometimes we just put a few objects on the floor--hide your eyes and somebody takes some away--you have to figure out how many are gone.
The Squirrelings have occasionally run into math blocks in other areas, but understanding addition and subtraction has never been a problem!
P.S. Here's a bonus for reading to the end: how do you say "Cuisenaire rods" in other languages?
French: les réglettes Cuisenaire
Spanish: las regletas Cuisenaire
Portuguese: as barras Cuisenaire
Italian: I regoli Cuisenaire
Polish: klockami Cuisenaire'a (Krakovianka, did I get that one right?)
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