That would seem to me to be a prerequisite for:

- good teaching
- writing good maths software about number

Now the ancient Greeks began their contributions to mathematics around the time that zero as an empty place indicator was coming into use in Babylonian mathematics. The Greeks however did not adopt a positional number system. It is worth thinking just how significant this fact is. How could the brilliant mathematical advances of the Greeks not see them adopt a number system with all the advantages that the Babylonian place-value system possessed? The real answer to this question is more subtle than the simple answer that we are about to give, but basically the Greek mathematical achievements were based on geometry. Although Euclid's Elements contains a book on number theory, it is based on geometry. In other words Greek mathematicians did not need to name their numbers since they worked with numbers as lengths of lines. Numbers which required to be named for records were used by merchants, not mathematicians, and hence no clever notation was needed.Well, my quest for number fundamentals led me to review several books. I ended up looking at about seven books using google books and amazon as starting points. Here are the two books I ended up deciding to buy:

Children Doing Mathematics by Terezinha Nunes and Peter Bryant (1996), 268pp (amazon link)

They are psychologists who are interested in children's reasoning ... Keith Devlin recommends this book for its treatment of multiplication, which I mentioned in an earlier blog (Multiplication is not repeated addition). This book is entirely devoted to number, which is what I wanted

Knowing and Teaching Elementary Mathematics by Liping Ma (2000), 166pp (amazon link)

This book asserts and documents the claim that maths is better taught in China than in the USA because Chinese teachers have a more profound understanding of maths knowledge. One thing that appeals to me here is that it contains concrete examples of a good way and a not so good way of teaching various maths concepts

I see a need a need to promote books about the fundamentals of learning. This has just come up again in discussion arising out of the IAEP (Its an Education Project) list, see this blog by Red Hat and Sugar Developer Greg DeKoenigsberg, promoting a book which is not about fundamentals but seems to be more of a lazy transfer of Clayton Christensen's concept of disruptive technology from the marketplace to learning. I left a comment on Greg's blog arguing that point

Summing up:

- Read books
- Study books about the fundamentals of knowledge and learning
- Good learning software (eg. logo, etoys, scratch) requires a process such as this