The more and more advanced compilers, languages, and frameworks we have that either automate tasks, or hide certain intricacies from us, do they ultimately make us dumber programmers?

Jim C gives the following analogy of a calculator:

Any one extract a square root lately, by using paper and pencil? We use a calculator or simple call a function in our favorite programming language. How many of us know what the calculator does or even the function?

So why don't we all just tell our kids to forget learning math, that's what we have calculators for? Wouldn't that inherently make our kids dumber?