I completely failed to follow your thoughts there. I can usually do simple arithmetic in my head - this does not extend to most division.
I'm not sure I can explain my mental process. It's really as if my brain subconsciously "sees" an easy method based on things I already know. In this case I "automatically" know that 7*20 is less than 175 and that 7*30 is more than 175. So I know right off the bat that the answer is 20-something. So I do the first part (7*20), then do the rest (175-140=35, so what is 35/7?). I solve a few simple problems to get my answer. As I think about it, it really is like long division.
It is only quite recently that I internalised the idea that 57 + 48 = (50 + 40) + (7 + 8). Let me tell you that realisation made mental arithmetic a whole lot easier! I never picked up the similar trick for division though.
There's nothing wrong with what you do, I just do that problem differently. I use two steps. 57 + 40=97, then 97 + 8=105. If I did it the pencil and paper way in my head, it would be pain in the ass (see below).
My question, though, demonstrated my point. People have different mental processes and perhaps what you could call
emotional responses to those processes. In your example, my method comes automatically and with ease. It
feels good/right. Your method feels "contrived" to me, and it "bothers" me that it takes three steps instead of two. That's not a dig on you or anyone else who does it that way because it's really a great way of doing it.
What do you find intimidating about it? You only have to hold a one number in your head to calculate this the 'traditional' way, divide 7 into 17 remember the answer and carry the remainder. No divide again and combine. Don't you picture it in your head like you were writing it down? That's how I do it anyway.
That is an excellent question. No, I do not picture anything in my head at all. Zero. Nada. My ex-wife was a visual person as well as a school teacher. We had many discussions on this subject when we were much younger.
It was a startling realization for me that people actually pictured things in their head like that. Likewise she was blown away that people did not visualize things like she did. She did the math much like you describe. She couldn't fathom how I could do it without "seeing" anything.
Each year in the classroom she would discuss our dog, Buck, a black Labrador retriever. She would ask the kids by a show of hands how many automatically created a mental picture of a black Lab. It was always about half the class. The other half had no mental image unless they deliberately took the effort to create one. For them it was more like the brain automatically recalled facts about black Labs. That's how it works for me.
Even as I sit here and try to picture the long division in my head, I find it very difficult to create and hold those images. I can do it, but it takes me several times as long as my "natural" method. And I have to shut my eyes.
This is why I have empathy for people who find the process intimidating.
