This is where new math excels - speed in computation.
I argued against manipulatives when my children were in school. My argument was not against manipulatives, per se; but that they hadn't yet learned the BASIS for using manipulatives, yet. And this is the biggest reason why US does so poorly on Math scores across the world.
Math is not stressed. Rote math learning is not allowed. Kids are fast tracked into newer math without understanding WHY the newer math works.
My .02
I was in elementary school in the sixties. I got a curious mix of old and new in math and reading - in my opinion, this is the way it should still be taught. The basic memorization came first, tables and phonics. Then the theory based newer systems were laid on top of that. I came out of that system understanding numbers and their relationship to one another on an intuitive level. My reading skills continued to grow. I believe the librarian pointed out two books to me in my sixth grade year that were the only ones of the thousands in the library that I had not checked out. Practice does help :D
However, I know what you mean about wiring. My daughter just cannot seem to grasp numbers this way. My son did. I know of other homeschoolers where the genders are reversed, so don’t take this as a “boys are better at math.”
My philosophy was different.
In my homeschool I used manipulatives exclusively until the child was 5. Before the age of 5 my children could add, subtract, multiply, and divide using manipulatives. At 5 I began doing traditional first grade math using manipulatives and pencil and paper. By the time the child was 9 they had all their math facts down cold! They could give any answer within 3 seconds. Then after age 9 I would use manipulatives as needed if they made an error in their work.
All three finished college Calculus III at our local community college by the age of 15.
I don't agree. I learned the ol' fashioned way and I have no trouble doing math in my head.
You see, as a person does the same task over and over, they automatically find short cuts. Calculating change, adding a column of numbers, long division, etc. New math takes you away from the foundation and takes you straight to the shortcuts. It's unnecessary and confusing for the kids.
Give them the foundation and, if they actually *do* math enough in the everyday lives, the shortcuts will come.