How regrouping in addition *actually works*

In kindergarten, or first grade, or whenever the heck we learned this (I don't remember), I think most of us were just taught to blindly "carry the one". There was no explanation as to what that actually meant; you were simply supposed to accept it as fact and go with it. Those who know me know that I hate black boxes, and the absolute bane of my existence is to be told that something is objective and indisputable fact, to be taken at face value without an explanation as to why. It irritates me to the point where it actually inhibits my overall understanding. When in the process of examining my own mathematical education, I believe that the gaps in my conceptual understanding started as early as this one. You end up memorizing steps, without actually understanding what's happening, and this inhibits you when you are expected to make your own original connections and insights (i.e. on an exam). And it sucks, because you feel like you're failing, when in reality you are simply not approaching it in the correct way. 

Unfortunately, it's a fact of life that most people who become elementary school teachers do not like mathematics. This is fine, because they have many other wonderful talents (elementary school was a great time for me, and I'm grateful for the teachers that I had), but it causes kids to grow up with insane mathematical deficiencies. I can't even count how many former GATE kids finish high school saying that they "hate math". For a short while, I was one of them, though I stubbornly persisted with it anyway and found that I really enjoyed math by the time I hit college. In most cases, I think this dislike is caused by conceptual gaps such as this one. Like an avalanche, conceptual gaps tend to pile up until you're taking college-level mathematics courses and struggling to get the grade you want. I'm sure that some kids were able to fill in the gaps on their own (my brother being an example), but I was not, and I did not understand it until doing my in-depth review of mathematics on Khan Academy. 

But enough of my B.S. (don't worry, it stands for bunny slippers); let's cut to the actual explanation. Will this be a good explanation? Honestly, IDK, but I know for sure that the Khan explanation is good. So, if this explanation sucks, check out their early math module. 

Explanation 

TL;DR You are adding groups of TENS and ONES separately. But if there are enough ONES to make a new group of TEN, then you HAVE to make a new group of ten, and adjust your addition accordingly. 

Let's add 35 + 6. 

As we can see, there are three tens and five ones in the number 35. By definition. 

When we add six ones to 35, we now have three tens and eleven ones. 

But, if we have more than ten ones, we need to regroup. So, we have four tens and 1 one: 

Which is 41, by definition. 

This is regrouping. Now let's do this symbolically: 

As we can see, what we do when we "carry the one" is separating groups of ones and tens, in order to simplify our addition. But your teacher probably just told you "carry the one" like it was a mindless, mechanical step, and for many of us, it absolutely was. This sucks. Until you understand it. And once you do, it's beautiful. 

I think that it would be prudent to offer some sort of financial incentive for those professionals working in STEM fields to spent some part of their week teaching kids math. Then, the ElEd types who hate it would be free of it forever, and the kids would grow up with less gaps in their understanding. Of course, we also have wonderful resources such as Khan Academy now, which I would have killed to have as a kid. So, who knows for sure.