What is a Math Concept?

I recently received an email from an educator in North Carolina encouraging me to look deeper into the meaning of the word "concepts". Below are snippets of the email. 

This email stemmed from a category on my website Sensible Mathematics 

https://sites.google.com/site/sensiblemathematics/activities-by-concepts. The writer felt the category should not read "Activities by Concepts". 

Because I understand there's a constant need for growth which happens when new information challenges my current understanding, I'm open to your perception, understanding and definition of a mathematical concept. Here's my current knowledge:

Lies My Teacher Told Me

I remember having to take a History course while attending the University of Florida (Go Gators). We were required to read a book titled Lies My Teacher Told Me. To me even the title of this book was so provocative, to even image that teachers would lie. As I read the book, it became more shocking to me as it discussed rtopics and ideas shared by teachers which were invalid or down right untrue. And a lot of this teacher misunderstandings came from dare I say, a textbook. 

If I were to rewrite this book based on math misconceptions I've heard from my students in the first 5 weeks of 7th grade I would include the following:

You cannot take a large amount from a smaller amount. 

A negative plus a negative ALWAYS equals a positive. 

When adding 36 + 47, 6+7 and 3+6. 

Adding means a number gets bigger, subtraction means a number gets smaller. 

When you add on a number line you ALWAYS go to the right. When you subtract on a number line you ALWAYS go to the left. 

And the one that takes the cake for me: when discussing absolute, the number comes out of jail and is positive. UGH!!

Rules can expire and tricks cannot be applied in various problematic situations. So is it truly worth the price students have to pay in misunderstandings?