I think I'd calculated combinations, but dubiously ignored where both were identical - that's where I ended up with 36.
OTOH do we memorise any of the x+1 combinations? I hope we don't, but perhaps we do. I genuinely can't say at this point. I was trying to work out how I processed sums such as 8+7 earlier, and concluded that so far I can tell, I work out the difference of one of those numbers from 10, subtract it from the other, then it's a very simple addition - ie that becomes 10+5. But I'm now unsure if that's what I do as a general rule, and am even less sure what other people may do.
Times table I vaguely recall learning by rote in formative school, but that's an awfully long time ago, and trying to self-analyse my mechanisms for multiplications is highly challenging. It feels like I try to move those back to multiples of 10 or 100, again, too.
I recall reading aeons ago that the only intuitive interface is the nipple - beyond that, everything is learned. So what makes for an intuitive or sensible mathematical representation of things is probably so arbitrary as to be pointless arguing about. It feels that the kinds of things we do, day to day, with numbers, that base-10 arabic number system is optimum, but that may simply be the lack of exposure to a better system.
As a parent of young children and former math teacher, yes, we do absolutely memorize those sums. It usually happens at an early enough age that the process of doing so evaporates early in life and becomes part of your base mental code. I spent a significant part of my teaching life teaching high school math for college students (and occasionally grade school math for college students) and there are very much people who never managed to get that memorization step completed. What's really fascinating is that it's an orthogonal skill to higher mathematics. I've seen students who needed to use a calculator to do 6+5 and yet managed to be able to solve algebraic problems. This is, I must add, uncommon, but it's less because one skill depends on the other but rather because the failure to gain the basic math skill leads to an unwillingness to try to gain the more abstract math skill.
I agree that Arabic numerals are better, but I did want to make the point that Roman numerals aren’t quite as bad as you might think having already learned Arabic numerals.
Your method for summing 8+7 is what I think I do for things like 7+4 (since I can visualize 7 as “three less than 10” and 4 as “one more than three” all in the same thought to reach 11), but for 7+8 my brain noticeably spits out 15 immediately and only a moment later does it actually do the processing you mention.
OTOH do we memorise any of the x+1 combinations? I hope we don't, but perhaps we do. I genuinely can't say at this point. I was trying to work out how I processed sums such as 8+7 earlier, and concluded that so far I can tell, I work out the difference of one of those numbers from 10, subtract it from the other, then it's a very simple addition - ie that becomes 10+5. But I'm now unsure if that's what I do as a general rule, and am even less sure what other people may do.
Times table I vaguely recall learning by rote in formative school, but that's an awfully long time ago, and trying to self-analyse my mechanisms for multiplications is highly challenging. It feels like I try to move those back to multiples of 10 or 100, again, too.
I recall reading aeons ago that the only intuitive interface is the nipple - beyond that, everything is learned. So what makes for an intuitive or sensible mathematical representation of things is probably so arbitrary as to be pointless arguing about. It feels that the kinds of things we do, day to day, with numbers, that base-10 arabic number system is optimum, but that may simply be the lack of exposure to a better system.