Education observation

I was sitting with my little girl as she was working on some math...we were doing subtraction with borrowing, and she'd learned from the book about the standard way of doing borrowing. So...me being me, I switched over to doing stuff in a more mental fashion and a less automatically. 796 takeaway 573 is properly done left to right. 796 takeaway 500 is 296, takeaway 70 is 226, takeaway 3 is 223. Great. But then the little brother came by to help. And little brother made a lot of noise. Real quick, kiddo's mental math skills dropped to zero.

Aretae's (AFAIK) original position on math:
If you can't keep enough stuff in your head at once, you cannot do math decently.
This requires 3 things.

1. Math chunking. 2x+1 needs to be a single memory item, not 2 items. Ditto 3/5. Hopefully, 256 also.
   
2. Memory hacking...using both short term (7 +/- 2), and sensory (~15 seconds). You need to keep a bunch of things in your head at the same time. If the steps needed in order to do step A are required memory items, you're screwed. If you take too long, you lose your sensory memory items as well.
3. Computation speed. If you can do 1 step in a half-second, you can float most of 30 steps in sensory memory.

Example: 3/4 + 5/6

• Note...if you have math chunking...that's 2 items in memory. Elsewise, it's 4.
• What's the first step...if you have to remember that (get the common denominator), that's 5 items in memory...which is in the range of what folks can remember.
• What is the common denominator for 4ths and 6ths? 12. I hope that didn't take too long, because now I have 6 items in memory if I needed to lookup step1.
• What step next? Hope this isn't occupying space.
• How many 12ths is 3/4? How do you figure that out? how many 12ths in each 4th? 3. then 3 4ths make how many 12ths? 9. Seems like that was 5 memory items.
• NOTE: if you weren't pretty fast, well-chunked, or using memory tricks, you're screwed. You can't keep all this in your head without tricks.
• Now, how many 12ths is 5/6? more steps to 10/12.
  
• And then finally 9/12 + 10/12 = 19/12, 1 7/12

Fundamentally, you have to decide whether to manage all this in your head or on paper. If you manage it on paper...you can outsource some of the memory. But I'm not entirely convinced that you don't also outsource some of the understanding onto the paper as well.

Fast computation, number chunking, and memory tricks. The path to good math skills is to get all this into your head, rather than trying to keep track of it on paper. Fact is we can't write fast enough to keep stuff in memory.

New hypothesis, discovered while writing this post:
Writing down math causes lower math understanding over time than doing mental math.