The first issue is illusory superiority. Almost everyone views themselves as smarter than everyone else. Obviously, this is impossible. Most people who think they are "good at math" are good at the math they use day-to-day. Everyone is good at that math. However, they are not good at complex equations. When people are given a new way of doing things, very rarely do they give it five minutes. They immediately disregard anything that does not fit their current world-view.
The second issue is that this is a very easy-to-understand solution that has been used by computers since, well, forever. And, if you ever purchased something with cash, the cashier most likely counted back your change.
It comes down to efficiency. For simple calculations, like 32-12, the "new" way is more complex, the "old fashion" way is much simpler. But, when you introduce number borrowing, the "old fashion" way rapidly becomes much more complicated.
As you can see, in my very scientific drawing, the complexity curve for the "new math" is flat.
Most of these people are upset with the implementation of the Common Core curriculum in their local schools. Instead of attacking the real problems of common core, they seem to be attacking the arithmetic, as if that somehow invalidates the entire Common Core program.
I can get behind the basic idea of Common Core. Much like the LDS Church's programs (no matter where you go to attend services, the same meeting formats and lesson materials are used), the original intent of the program is to create a unified basic structure for learning fundamental principles.
One of the reasons behind Common Core is the ever-increasing number of student transfers. Most of those transfers are due to things such as the economy and the divorce rate. Students go from one school that is teaching them to solve problems one way, to another school that teaches them to solve problems in a different way. This creates confusion and factors into declining test scores. Common Core attempts to solve these problems in a noble way.
However, Common Core also includes an entire array of standardized tests, and there are many problems with standardized testing. There are movements in many countries that are pursuing dismantling standardized testing in favor of personalized education. Standardized testing creates an environment for students where being wrong is one of the worst things that can happen. Divergent thinking is being educated out of our children and replaced with groupthink.
Sir Ken Robinson once said, “The fact is that given the challenges we face, education doesn't need to be reformed -- it needs to be transformed. The key to this transformation is not to standardize education, but to personalize it, to build achievement on discovering the individual talents of each child, to put students in an environment where they want to learn and where they can naturally discover their true passions.”
The next time you want to attack Common Core, attack it based on the actual problems it presents. Not the math you don't want to understand.