Hard to explain. My wife and I play a lot of darts. Not always Cricket, often 501 (you must achieve exactly 501 points without going over, it’s kinda hard).
We also play Cribbage. Also a lot of counting.
My wife is a real estate appraiser (or was until she retired last week). So she deals with a lot of numbers. Big ones. I’m a programmer, so I’m logic.
But we count different. We have different systems that we use in our heads. I can tell. So can she. I can’t quite put my finger on it, but we come up with the same results with different ‘systems’.
How would that work? Are you saying that there are times when you and your wife are both at number x and you don’t agree on what the next number above x is?
I look at that and see 54. 9+11 = 20+20 = 40+14 = 54. Easy peasy.
My wife will do them in order.
9+20 = 29+14 = 43+11 = 54 Ya goota carry more numbers.
My wife and a coworker also noticed that they count differently. So they had a ‘count off’ Like 10 sets of numbers. They both got it exactly correct.
I noticed this playing darts. I’ll shoot, walk to the board and count. My wife now knows not to do this but she would also walk to the board with me and start counting. Out loud. It confused the shit out of me, as I was counting in a differnet order.
My job involved a lot of mental arithmetic with 1, 2, and sometimes low 3- digit numbers.
Like the OP, I’d often resequence them to get simpler intermediate results. I’d also round them, add them, then back out the rounding.
Lots of work arithmetic is approximate. Round, add or multiply, and that’s close enough; no need to reverse the rounding. Eg. 38x19? Call it 40x20 = 800. Maybe back it off WAG 50 if I thought 750 vs 800 made a significant difference in whatever it was.
There are plenty of gymnastics you can use for this stuff. I find it fascinating. I see 12 objects, I see 2x6=12. Or 3x4=12. I do a lot of grouping I guess. I think my wife would go, 1,2,3,4,5,6,7… Where I would immediately group them.
I think that I do something like what the OP describes. When faced with several numbers to add, I will do what I call “picking out the tens.”
So let’s say that I have to add 8+5+4+2+1+7.
Okay, there’s an 8 and a 2. That’s 10. Then there’s a 5 and a 4 and 1. That’s another 10, so now we’re at 20. And the only number left is 7, so the answer is 27. Easy.
I can remember referring to this on Facebook once, and several people had no idea what I was talking about, and seemed to find it very confusing.
No, no, no, you’re doing the tenner groupings wrong. Start from the far end, it is 7 + 1 + 2 that makes ten, and 4 + 5 also make ten if you borrow one from 8, so that 7 remain, therefore, as you rightly guessed, 27.
But sometimes it is better to group in 15s or 20s or 50s. It depends. And sometimes it is better to directly multiply. Division is fun too.
Yes, I add a list of numbers by rounding them to the nearest ten and keeping a run-off number so that, when I reach the end, all I have to do is add/subtract the run-off number. I then have an exact answer far more quickly than I would using standard addition.
I don’t think it’s a matter of counting, exactly. I’d guess that everyone does that more or less the same way.
Mental arithmetic is a whole other issue, though. I suspect there is lots of variance there. I was brought up to do arithmetic the ‘standard’ way, but over the years I have evolved a lot of shortcut and estimation methods that I use habitually these days?
Likewise. Lots of ‘short cuts’ to get to the final answer. I don’t estimate in Cribbage or Darts. Those are real numbers. But, I add them differently than my wife will. And 1 point often determines the winner.
How much top soil or gravel you need is a bit different, way too many variables.
It’s different than ‘Pop’ or ‘Soda’ or give me a ‘Coke’ “What kind?” (which is totally screwed up). This is not regional I don’t think.
I do think that it is a good example of how we are wired differently. And that life, jobs/careers do push you to think in a certain way.
For example - I was a framer (houses) many decades ago. We would make notations, and work with the numbers on the wood that we where about to install. The wood grain often made it hard to do circles. My 2 is still a Z. My 8 is a Z with a backslash through it. A Z is easy to write on wood, a 2 not so much.
If I’m reading a book that’s 332 pages long but I’m only on page 187 and I want to know how many pages I’ve got left to read, I don’t mentally subtract 187 from 332 directly. I don’t say to myself 12 minus 7 is 5, and I’ve got to borrow 1 from 3 to get 2. I don’t then say 12 minus 8 is 4 and borrow 1 from 3 to get 2. I don’t then subtract 1 from 2 to get 1. All this makes the answer that I’ve got 145 pages left. That’s how you learn to do in school.
I say to myself that I’ve got 13 pages left before page 200. I say that 332 is 132 pages more than 200 pages. So I’ve got 132 plus 13 pages left which is 145 pages left. For any other arithmetic problem, I just put it into Google on the computer I’m on or on the calculator I’m holding.
Incidentally, there are languages and cultures with no numbers or only small numbers.
You don’t have to actually mentally carry any numbers at all IF you have certain mathematical facts regarding addition memorized by rote. It’s similar to how many people have the multiplication tables through 12x12 memorized – you don’t have to replicate pen-and-paper multiplication in your head to recall that 12 x 11 = 132, or 12 x 12 = 144.
To make that kind of “in order” addition easier to execute in your head, you have to have the results of the carrying memorized by rote:
So, the same way that, in multiplication tables, 12x12=144 is kind a singular “memory unit” that gets recalled as a block of info (as opposed to a sequence of calculations done on pen and paper), the “8, 7, 5” in the first equation above are a singular memory unit for addition. And similar for the other two equations.
That “+1” part in the right side of the equation is also memorized to the point of not requiring active recall.
Yes. The only ones I know of are a couple of primitive tribes in the Amazon.
Our brains are hard-wired to recognize that we have none of something, one, two, maybe three or four (I forget exactly where it ends), “a few”, and “many”. So even these primitive tribes have words for these concepts.
These tribes get along just fine without a formal number system. You don’t really need numbers until you start getting into more complex trade.
Zero as a number took a surprisingly long time to develop. In ancient times, zero wasn’t a number, it was a lack of numbers. In other words, it was a concept, not a number.
As a child, whenever there was occasion to do a countdown, I always counted down from 8 to 1, likely because I’d heard it done that way in some song or TV show. When I came to understand that the rest of the world counted down from 10 to 1, it felt profoundly jarring - like getting to the top of a staircase and stepping on a stair that’s not there.