Arithmetic by Paul Lockhart.
4/5 I enjoyed learning about the history here. Lost interest somewhat during fractions and multiplication because it seemed to focus only on how it worked and supported place value rather than on the history for those. Exercises were an interesting way to explore the concepts.
Most interesting things
- Roman numerals didn't use prefixing until being used as convenient labels in the 1300s, counting was merely appending and collapsing values
- How the abacus and tableaus relate to physical representations without place value. I feel however though that place value really just emerges from these anyway, you still increment values in place, it's just easier to move them around and there's no cap on the value in a particular place.
- Physical representations of numbers vs symbol memorization. It seems that for basic arithmetic this would be sufficient and helpful most of the time. Multiplication and division feel like they're used as explainations for a difficiency here, but it feels very possible to implement those physically as well.