John Holt's Learning All the Time offers a compelling account of the ways that some conventional instructional approaches to math can hinder children's understanding of the relationships between numbers; he describes alternative ways to foster children's sense of the connections and relationships between numbers rather than focusing on "math facts." For example, you can explore all the different ways to make six rather than concentrating only on 3+3. Math is all around us, and children don't necessarily need to master addition before subtraction or even multiplication.
So far, N.'s exploration of numbers has naturally followed Holt's account and it's been fascinating to watch. His interest in numbers has arisen out of his play and his passions. For example, when he was 4, he learned that steam train engines are described by the number of bogey, driving, and trailing wheels they have. So a 2-6-4 has two bogey wheels, six driving wheels, and four trailing wheels going from the front of the engine to the back. But when you look at a 2-6-4 from the side in illustrations (in the zillions of train books we own!) you see one bogey wheel, three driving wheels, and two trailing wheels. So to identify the train properly, you have to double what you see in the picture. I don't recall us explaining this to N. (because frankly I had no idea what this chain of numbers was that was always listed with the make and model of a train), but he looked at a lot of books and apparently from pondering the disparity between the identifying label and the picture, he figured out how to multiply by two. And when he figured this out, he took immense pleasure in doing this operation. Whenever he looks at a picture of a train he says what its wheel configuration is. So now he has a concept of multiplication that we can refer to when we talk about other ways that numbers can be put together or split apart.
Whenever questions about numbers come up in conversation (and they do all the time), we try to build on them and extend them. So, if N. asks us what five and seven make, we'll say 12, and we might follow up by asking him what seven and five make. Usually he likes to think about it and respond; if he doesn't we'll answer our own question. If he's wrong, we'll just say pleasantly what the answer is. We're not quizzing him, but engaging in thinking with him about numbers.
Other ways that I've observed N.'s math skills developing include his extensive play with blocks and legos. Not only is this kind of manipulative play crucial for understanding quantity and number stability, but it also gives him lots of opportunity to add and subtract, to create symmetry and asymmetry, to consider the relationships of shapes to each other (proto-geometry).
We also do a lot of cooking and baking together, which of course provides a great opportunity to play with fractions. I've enjoyed watching his slow grasp of "half," "quarter," and "third" (which has been especially tough). It's been really interesting for me to see how hard these concepts can be and so it seems important to provide lots of opportunities to talk about them, explore them, and get them wrong.
The two dominant modes of N.'s math development so far have thus been through conversation and manipulation. But N. has also said to Tim "Let's do some math, Dad," or "Let's do numbers," so they've done some practice with written numbers. Here are some examples:
Even and odd numbers:
Writing numbers in different fonts!
N. is by no means a math prodigy, and what he's done on his own and our numbers work with him is not anything out of the ordinary. I've tried to describe here both that he has an innate interest in numbers and in other mathematical concepts and that we've tried to build on that in positive, no-stress, non-coercive ways.