Good primer on K-3 math instruction. Mostly not new to me, but helpful if you're just starting math instruction: avoid testing if possible (although you can test to find gaps or build confidence), definitely don't teach to a test, embrace the advantages instead of trying to make homeschooling like institutional schooling, use real life to teach math: games, kitchen, shopping trips, etc.
New to me was the three modes of math thinking:
1. Manipulative
- Physical manipulation, like counting fingers or blocks
- Only mode until age 6 or 7
- Greatest source of math anxiety is an underdeveloped ability at doing manipulative math
2. Mental Image
- Mental manipulation, like counting blocks in your head
- Develops as child uses physical manipulatives
- Eventually it will be too slow to count on their fingers, and they will realize they can do it faster in their head
3. Abstract
- Abstract manipulation, like 2 + 2 without picturing "2" of anything
- Piaget says abstract thinking doesn't develop until age 12
- Delay teaching symbols until child is ready
I've read that claim that children don't develop abstract thinking until 12. Obviously a younger child can solve 2 + 2, but maybe they're doing mental manipulation or memorizing without understanding? But children do all sorts of abstract thinking before 12, like reading or using a calendar or a telling time on a clock. Is that kind of abstract thinking somehow different than abstract math?