In recent weeks, I have been trying to take a more hands-off approach to the boys' learning in an effort to see what directions it takes autonomously. The results are often surprising. Our third son (who recently turned 7) has come out with a couple of mathematical insights completely spontaneously this week, which I felt ought to be documented here. They are insightful to the discussion about maths education. He said, "Mum, if you add two odd numbers together, you always get an even number" and proceeded to demonstrate how this was so, clearly enjoying the investigation in his own mind. On a separate occasion, he said, "Mum, you can divide an odd number of sweets equally if you cut one in half, so if you shared 11 sweets, it would be 5 and a half each" and he went on to give other examples.

I have also noticed him figuring out time in his own mind. I have worried in the past about the boys' struggling to read an analog clock, but this complex skill actually requires many threads of knowledge to come together and to be applied ... fractions, 5 times table, a concept of minutes and hours. This happened quite spontaneously for our second son - perhaps later than school would have liked, but in his own time - as his understanding of time developed. Now his brother is doing the same, looking at the numbers and fitting them into his understanding of his day - not the other way round. My concern is that children are not given the time and space to form the concepts before we are measuring their ability to look at the numbers. And even if an understanding of the numbers seems apparent, is it real understanding if the underlying concepts are missing?