How do they learn math with unschooling?
They learn math from using math. Games are math. And far more complex math than is taught in school. Game, especially video games, use math displaying charts and graphs and maps and percentages and so on.
They learn math because people around them speak math and it's a handy and fascinating tool. Speaking math means figuring out the answers to questions out loud.:
"How long until Daddy gets home?"
"Well, it's 3 now and Daddy gets home at 6 so" counting on fingers "3 to 4 is one hour, 4 to 5 is another, 5 to 6 is another. So 3 hours." And I usually converted that into number of favorite half hour shows since that was a familiar chunk of time and it immerses them in fractions and multiplication too. "There are 2 Magic Schoolbuses in an hour, so 3 2's is 6 Magic Schoolbuses until he gets home."
And, "How long until Christmas?"
"Well, it's April now and Christmas is in December." counting on fingers "April, May, June, July, August, September, October, November, December, so 9 months. Each month has 4 weeks. And 9 4's is 36." (Or you could do days. Or be more exact as they get older and as Christmas gets nearer.)
And, "I've got 58 cents and I found a quarter. Do I have enough for the $1 baseball card pack yet?"
"Well, 2 from the 25 gets the 58 up to 60, so we've got 60 + 20 + 3 or 83 cents. Which is 7 to get the 83 up to 90 and 10 to get the 90 up to 100, so you need 17 more cents." (That's one way of breaking up the problem. There's oodles of ways. I think it's vitally important to avoid pencil and paper math -- unless they like it of course! Break problems up in your head so you can solve them out loud. That way kids can see how numbers work.)
They don't need to fully grasp it, just get the gist of it. Just as they don't understand every word we use when we speak to them. They just get the sense of what we're saying from the words they do understand. But the presence of the words (and the math concepts) they don't fully understand are just as important as the words they do understand because they're in the process of understanding them. They can't acquire new words unless new words are available.
Now, this thing about no paper .... hmmmmm. I always get out the paper (especially for division) mainly because I lose my place and forget the numbers I have already come up with.
That's why you need to simplify the problem so you don't have all those numbers floating around. How often is a correct-to-the-umpteenth-decimal-place answer necessary? When it is, I use a calculator. When it isn't -- and it usually isn't -- I estimate or simplify.
(The one everyday exception to not needing exact answers that I can think of is trying to figure out price per unit at the grocery store to compare two things. Our grocery store posts price for unit for regular prices but often doesn't for sale prices :-/)
Estimation is a hugely valuable tool. And yet it's given a chapter in math textbooks. We were told that we should always estimate but in practice it was engraved on our brains that what was important is an answer to the umpteenth decimal place. So in school it seemed why bother? And being an anal person, I want those umpteenth decimal points ;-) So losing the paper has forced me to let go of that anal need to great benefit.
Addition is easy enough to simplify since you just move numbers from one to the other or add some in and remember to subtract it off later. (17+29 is the same as 16+30.)
Subtraction is a little trickier but I've usually approached it as "how far away" so 127-58 is how far is 58 from 127. Adding 2 to 58 gets you to 60, 40 gets you to 100, 20 gets you to 120 and then add 7, so 69.
Multiplication can be grouped and done a bit at a time. That's where the fact that multiplication is repeated addition becomes obvious and useful. If it were 52x17 I might rough estimate it's going to be closer to 1000 (50x20) than 500 (50x10). If I needed exact I'd do 50x10 + 50x7 then 2x17 (or break it down too if that was too tough). So 500+350+30+4 which is 884.
Division is trickier but it's repeated subtraction. It's mostly dividing things in half and quarters so you can take half and then half again. They can be broken down too. 127/5 is 100/5 + 27/5. If it were trickier I'd probably add/multiply until I got up to the number. So 127/17 I'd keep adding 17's until I got close.
I think the motivator is having a child standing in front of you who wants the answer right now who will probably give up if you have to go searching for a calculator or pencil and paper ;-) It forces you to come up with quick and dirty ways to solve problems :-) The side benefit is that they get to see how numbers work if you talk through the process out loud.