Sorry, I strongly disagree.
I did memorize names of numbers, but that is not essential in any way to doing or understanding math, and I can remember a time where I understood addition but did not fully understand how names of numbers work (I remember, when I was six, playing with a friend at counting up high, and we came up with some ridiculous names for high numbers because we didn't understand decimal very well yet).
Addition is a thing you do on matchsticks, or fingers, or eggs, or whatever objects you're thinking about. It's merging two groups and then counting the resulting group. This is how I learned addition works (plus the invariant that you will get the same result no matter what kind of object you happen to work with). Counting up and down is one method that I learned, but I learned it by understanding how and why it obviously works, which means I had the ability to generate variants - instead of 2+8=3+7=... I can do 8+2=9+1=..., or I can add ten at a time, etc'.
Same goes for multiplication. I remember the very simple conversation where I was taught multiplication. "Mom, what is multiplication?" "It's addition again and again, for example 4x3 is 3+3+3". That's it, from that point on I understood (integer) multiplication, and could e.g. wonder myself at why people claim that xy=yx and convince myself that it makes sense, and explore and learn faster ways to calculate it while understanding how they fit in the world and what they mean. (An exception is long multiplication, which I was taught as a method one day and was simple enough that I could memorize it and it was many years before I was comfortable enough with math that whenever I did it it was obvious to me why what I'm doing here calculates exactly multiplication. Long division is a more complex method: it was taught to me twice by my parents, twice again in the slightly harder polynomial variant by university textbooks, and yet I still don't have it memorized because I never bothered to figure out how it works nor to practice enough that I understand it).
I never in my life had an ability to add 2+2 while not understanding what + means. I did for half an hour have the same for long division (kinda... I did understand what division means, just not how the method accomplishes it) and then forgot. All the math I remember, I was taught in the correct order.
edit: a good test for whether I understood a method or just memorized it would be, if there's a step I'm not sure I remember correctly, whether I can tell which variation has to be the correct one. For example, in long multiplication, if I remembered each line has to be indented one place more to the right or left but wasn't sure which, since I understand it, I can easily tell that it has to be the left because this accomplishes the goal of multiplying it by 10, which we need to do because we had x0 and treated it as x.