My seven-year-old daughter has been bugging me to play math games with her for the past couple of days. I know...she shouldn't have to ask me twice, but we've been busy with recovering from a trip to the beach plus a bunch of home improvement projects. So last night I took some time to get out the "What's In A Number?" card game, and we played for quite a while. And it was great! I remembered why I created this game--why I think it's so important.
The cards display the visual factorizations of the numbers 1 to 24--in other words, pictorial representations that show dots grouped into the factors of each number--two card for each number. We had played in the past, but it had been a while, so she took some time to get familiar with the cards again. We took out our number rods and she built some of the numbers on the cards: for the twelve card she put three four-rods in a row and noticed that it came up to the twelve mark on the rod track. Then I took it one step further and gave her four three-rods and she checked to see if they were the same amount. I could see the wheels turning in her head as, instead of counting by ones as she usually does, she began counting the dots on the cards by twos when shown in pairs, and was even groping toward counting by threes in cards like fifteen (five triangles of three). When she was looking at sixteen, I handed her four four-rods, and instead of putting them in the rod track to check their length, I asked her to make a 4x4 square out of them, and she saw how that related to the picture on the card.
Then, before this could get old, we spread all the cards face-down and played Memory. She likes the game of Memory, and is fairly good at it. I think using the factor cards is more difficult than playing it with pictures (like our Dr. Seuss Memory game), so we make it easier by allowing six cards to be turned over on each turn--you can collect up to two sets of matches per turn. For her, this is just the right level of difficulty to keep her interested but not frustrated.
Tomorrow we'll play another game that uses these cards, the "What's In A Number?" game, where you are scanning your cards looking for factors that match up with the ones on a target card. Good fun!
So why does this matter?
Last year in first grade my daughter would come home from school with pages of addition and subtraction flash cards that I was supposed to cut out and quiz her on (which I refused to do). There are a bunch of problems with this approach, but here are two major ones.
First, showing a child symbolic representations of numbers (numerals written on paper) does nothing to solidify her sense of what the number actually IS--not just how large it is, but what tricks it plays (prime and composite, square and triangle numbers) and what it cannot do (odd numbers cannot be divided in half without chopping up a unit). Playing around with numbers in the real world, using as many of the five senses as possible (when is the last time you and your child made a 2 1/2 batch of your favorite cookies?) allows your child the opportunity to shake hands with the numbers, to really get to know them. Which is invaluable when it's time to start chopping them up, doubling and tripling them, etc.
Second, schoolish math has as the main goal (not in name, but in practice) good scores on tests. It doesn't encourage children to explore the world of number and make meaningful connections. Schoolish math is generally accompanied with pressure--adult-world pressure--to get the "right answer" and get it quickly. A 10-year-old child I know, who was raised on schoolish math, has very little sense of what the answer to an arithmetic problem should be; instead she wants to grab a pencil and write down the problem in the formats the teachers have taught her that "ought" to yield the right answer. Whether or not she actually gets the right answer is immaterial, since she really has no idea WHY it is right or wrong. In a world where calculators are ubiquitous, what we need are people who know how to think about math, who understand what is going on in a math problem.
Brain science has come a long way (though there's still a long way to go). One of the things I find fascinating is that there isn't a "math part of your brain." Math requires integration of multiple brain areas to solve even fairly simple problems. First there is what is seen and heard: the symbol, the written word, and the sound of the spoken name of the number--all are processed in different parts of the brain. The short-term memory must be activated to keep a problem in mind, and the long-term memory must be triggered to retrieve data related to the problem. The spatial and logic areas are very active in mathematical problem-solving. It's well known that "neurons that fire together, wire together"--in other words, you form math problem-solving highways (neural networks) in your brain when you are actively engaged in thinking about math.
But how to get all those neurons actively engaged? ("Anyone? Anyone?") No, not through droning on about it...just play! As you use numbers to do things that interest you (like trying to win games) your brain is furiously processing all the information that's coming at you, storing it away with a post-it note that says "Pay Attention To This! Important!"
So should children be plagued with flash cards? Maybe, but not at first. It is far more important that a child have opportunities to build up a "feel" for what each number is, so that they'll know how it will behave when it gets together to play with other numbers. Children need to understand arithmetic before being required to memorize anything. I am flabbergasted that the school didn't begin teaching addition by looking at what odds and evens do to each other (an odd plus an odd is an even, and even plus an even is an even, an odd plus an even is an odd--ALWAYS--so don't try to tell me that 8 plus 5 is 14--it NEVER can be). Having these types of discussion/demonstrations builds up a child's understanding of numbers.
Eventually, understanding and memorizing come together into a single "knowing." "Do you know your multiplications facts all the way up to the 12's?" can be answered with a solid "Yes," since even if you momentarily forget what 9 x 12 is, you have several quick strategies for coming up with it. And you have a good estimation of what it should be, so that if your strategy is a fail, you know it instantly.
And that's what's required to have a good number sense. If your child doesn't have one, go back to the basics and get one.