Story Problems First
There are books, and books, and websites, full of problems designed for young children. Problems that supposedly make math interesting. Problems that supposedly teach concepts.
I don't like any of them.
They ain't got no soul. They depress me.
The people that make up these problems, do they actually enjoy math?
I worry I am just an egomaniac who likes only his own problems. Maybe you have the same worry. But hear my argument.
Story problems are a natural use of math. Take away the story, and you have an abstract representation. For example, suppose a box contains a girl and a linzoony. There are 5 arms in the box. How many arms does a linzoony have? The abstract mathematical representation would be 5 - 2 = ?
Standard math teaching is to first teach that 5 - 2 = 3. Then this "skill" is applied to the story problems. It is well-accepted that children have difficulty with story problems. (The "key word" method is designed to allow students to solve the problem without understanding them. It is a testimony to the failure of current practices.)
The abstract representation encourages memorization, not learning of math concepts.
I start with story problems. Even young children seem to have no difficulty with them. I think they are actually easier for children. They also encourage understanding and learning concepts.
Psychological experiments usually find that people can solve problems better in natural settings than artificial settings.
The abstract representation then can come out of the understanding built up from problems. For example, you want children to understand what subtraction is before you teach them that 5 - 2 = 3; you want them to eventually memorize 5 - 2 = 3, but when they first see it, they should connect it with meaningful real-life operations.
So, you should start with story problems because that is a more effective way to teach mathematics.
I like math. I think children can like math. I have no problem getting children to like math. I cannot make math the most enjoyable thing in their life. But I can help them find the enjoyment in math.
And who can enjoy abstract mathematics? At some point, at some level of development and maturity, it can be enjoyable. But not at the start.
Algebra 5 = 2 + x; 7 = x + y. Solve for y.
Story problem. One box contains a girl and a linzoony. That box contains 5 arms. A second box contains a linzoony and a katszoony. That box contains 7 arms. Draw a picture of a katszoony.
I am teaching a second grader who likes faeries. I try to make all of my story problems involve her, her dolls, or faeries. Bob sez: Try to reformulate your problem as a story problems involving faeries (or whatever). If you can't, think about whether your want to give that problem to a young child.
I went to a website and tried to find a random problem. It was 0.6 - 0.2 = 0.4. Should children learn to solve this? Eventually. Is this interesting math? I don't think so.
Can I think of an equivalent story problem? Someone cut an apple into 10 equal parts. The faerie named Shimmer has 6 of these tenths. Shimmer then gives two of the tenths to Shine. How many tenths does Shimmer now have? Obviously, this is a simple subtraction problem. But so is the abstract problem.
The abstract problem has a difficult representation of tenths. That should be taught, eventually. And of course there is a very difficult underlying concept of decimals and what they mean. But the abstract problem doesn't teach any of that.
The abstract problem makes it more difficult to solve the problem meaningfully, because of the difficult terminology. It leads children to solve the problem without understanding.
I know, the reality is that sooner or later you have to teach your child to solve problems in mathematics that are completely abstract. You can try sooner if you want. But that is a tough way to teach concepts. It will probably lead to just memorization and a student who cannot solve story problems very well.
Story problems are considered hard by students, but if you are learning concepts and understanding math, they are not difficult -- they are just using what you learned. And for young children, they are more natural -- they are going to be more interesting, and they are going to make it easier for you to teach concepts.
Oh yeah, one more thing -- when I follow my own advice, my problems get better.