' Dr Ell's Math Blog: December 2009

Tuesday, December 8, 2009

In the Bag

     There is a useful game described in Patricia Davidson's Idea Book for Cuisenaire Rods at the Primary Level. A child, with his hands behind his back, is given a rod and asked to guess the color. This gives the little person a very tactile sense of the relative sizes of the rods. We found that identifying the very large and very small rods was not so difficult, but distinguishing dark green from black was much harder. The game was fun and helpful in developing real familiarity with the rods.
     As a variation on this theme, we made a game of placing a bunch of rods of all sizes in an opaque cloth bag. The first version of our game required the player to grab a rod (unseen) and predict the color before she removed it from the bag. When this became too 'easy', the mission became finding a specific color rod in the bag and pulling it out. Another variation had the player building a staircase and then a reverse staircase-- in other words, rods of ten different sizes had to be pulled out of the bag in order from smallest to largest then from largest to smallest. This is easier if there are only ten rods in the bag (ok as a beginning version) but harder when there are three or four or five of each color in the bag.

     These sort of tactile games are more important for some children than for others, but I have found them useful all around. Children with learning disabilities or deprivations of various sorts seem to especially benefit from the exercise. I once worked with children so culturally deprived they couldn't, at ages five and seven, recognize themselves in a photograph or understand two dimensional representations so as to make sense of a story book. They were enormously helped by a French game called Tactilo which I can't seem to find anywhere now. It was by Fernand Nathan and consisted of beautiful little wooden shapes-- cube, sphere, egg, cone, mallet, bowl, mushroom,etc and cards with matching two dimensional pictures. The object was to reach into a little silk bag and pull out objects to match the flat pictures. It's easy to take skills for granted that little ones need to learn in one way or another in order to move forward in math. And it's also why it's important to allow for plenty of block play, math games, spatial relationship and pattern learning to take place before "jumping the gun" into formal numerical instruction.
     When I was, recently, discussing the 'bag' game for Cuisenaire rods with my daughter, she made the suggestion that the game could be adapted to Pattern Blocks as well for the purpose of learning the shapes by feel. Sounds like a great idea to me.

Wednesday, December 2, 2009

What Fits?

     This is a Cuisenaire Rods game that lets your child begin to 'see' or 'feel' or 'sense' math relationships. The object is to start with a given rod, say the orange one, and 'fit' other rods to it.  Two yellows fit. A dark green and a purple fit, too. Ten white rods fit. And so on.

     No numbers need to be assigned, but they will come eventually. An orange rod is ten units long and the rods that 'fit' are the numbers that add up to ten. I think it's important to just go with colors and size and feel for quite a long time and let the game be open-ended. When arithmetic finally does come up it will seem perfectly reasonable that 5 + 5 = 10 because "of course" yellow and yellow fit orange. I find it helps some children to verbalize the relationships just that way: "Look, Mom-- brown and red make orange"  or "Green and red fit yellow."
     Cuisenaire sells a little frame with their basic sets that make 'fitting' the rods easier for little people. This frame is especially useful if the game is amplified to include more than two pieces in a 'fit' and if the 'fits' are stacked up into a pattern or design. These 'rugs' are a lot of fun to create. Some pictures of individual fits as well as rugs follow.

     As children get older and more experienced, the game can be amplified to include 'fits' to double length rods. For instance, the question could be asked, 'What fits two brown rods?' Fits can also be more challenging if the player is limited to a single color answer. For orange this would require two yellow rods or five red rods. For 'odd' size rods, like yellow, there is no single color answer. That problem is solved if a single white rod is permitted when needed. In that case, yellow could be fitted with two red rods and a white.

Tuesday, December 1, 2009

Mathland Weddings

     This game introduces the ideas of even and odd. The photos here use white Cuisenaire rods, but literally any kind of manipulative will work as well. Note that the game does not require that the child already understand numbers in the formal sense.
     Given a handful of blocks, demonstrate pairing until all the pieces are used up. If everything pairs, the group is ‘even’. If one is left over, the group is ‘odd’. Five year old Rebekah suggested that the ‘extra’ block was the ‘priest’ performing the wedding ceremony. So, if there’s a ‘priest’, the group is odd. No ‘priest’ means the group is even.                                       
This group is even. There is no left-over block. The second group is odd. There is a 'priest' to perform the ceremony. The third group is also odd.

     For children who can count to five and can understand the notion that ’five’ is the number of these blocks, you can introduce the idea that ‘five’ is always an odd number. Your little person can experiment with a variety of objects to verify that five toy cars, after the pairing process, will still leave one car left over. The same process can be used to examine ‘four’ which is always even and ‘three’ or any other number. The number ‘one’ might be a challenge but makes for interesting discussion. In Rebekah’s case, the reasoning went something like this: “Nobody married nobody but there was still a priest in the church so it was odd.” We didn’t bother with ‘zero’ at that time, but if we had, the discussion would probably have come to a conclusion something like this: “It’s always odd if there’s a priest and if there’s no priest it’s even. So if nobody marries nobody and there’s no priest either, then its even.”

This is a more challenging group because of it's size.
     There are many other possible variations on this game. The priest might be a rabbi or preacher instead. Or, since little boys don’t always get into the ‘wedding’ thing, the whole game can be turned into making pairs of shoes or boots for imaginary people, soldiers, whatever. When the blocks all pair, the group is even. A left-over boot means the group is odd.

The Inside of the Inside: a Cut and Paste Project

     This is an activity for children who have enough dexterity to use scissors and enjoy craft projects. The objective is to take a page or part of a page from an old magazine and cut out a piece then cut out a second piece from the new piece. (Pictures follow.) The child can then create a collage with the pieces. The project can be either abstract– just a colored piece of paper or a page with an abstract design, or it can be fairly concrete– an object within an object.
     The point of the exercise may not be obvious, but it helps a young person learn about wholes and parts, relative locations, and that the pieces are smaller than the original whole– ‘the inside is smaller than the outside’ idea. It’s a project that blends math and art and also lends itself to further elaboration for little people that enjoy the activity. For instance, the number of insides can increase or the cut-out shapes can take on more complexity.
Step 1: Collect a piece of a magazine page, a sheet of plain paper, scissors and glue.

Step 2: Cut a piece from the magazine page. The page can be creased to make it easier to start the cutting process.

Step 3: Cut a smaller piece from the first cut-out.

Last Step: Paste the pieces on the paper sheet.

Here are a couple of examples of more 'concrete' designs: 

A note on the conservation of matter: Your little person can re-assemble the pieces before the glue-down step to begin to understand that cutting and taking out pieces doesn’t result in ‘more’ or ‘bigger’ except in the sense of ’spread apart’. This idea is much less obvious than you might think.
     More pieces, more complex design: