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Honestly I am at a loss. I've even looked for workbooks to explain it but none of them do. I have sat down and tried to explain it to dd a few times and it leaves us both frustrated.
Is there a good website that explains it?
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Are you talking about addition and subtraction? You could try using a visual. There are workbooks as I have some but I'm not sure exactly what you're talking about.
Here is a description of how I showed it to my youngest. It's important that kids have a pretty good understanding of place value first, of course. And it would also be good if they were familiar with the manipulative model for a while first. We live in Canada where we have convenient $1 coins for the 100 placevalue. You could substitute bills, or if those don't have the right feel, use buttons or "make" some dollar coins yourself. But really, in the earliest stages you could just use pennies and dimes for 1's and 10's.
What I hope is clear from this explanation is that I didn't introduce regrouping as an algorithm to be mastered, but rather let my child discover the need for some sort of reassignment of place value by getting her really familiar with multidigit addition without regrouping and then eventually letting her encounter a problem where the sum in the 1's column was larger than 9. She *knew* that 10 pennies made a dime, so she saw her way to the regrouping solution on her own. The sense of the algorithm was introduced in this discoveryoriented way before the algorithm itself. Only after she'd discovered the sense of the regrouping did we translate it into written digits  by working problems in parallel with manipulatives and digits.
We have always used the term "regrouping" (i.e. creating or breaking up a unit of higher place value) rather than the terms "carrying" and "borrowing" which aren't really accurate. (After all, if you "borrow" something it is "owed" back, but that's not the case with regrouping for subtraction. No value is lost from the number.)
Hope this helps.
Miranda
Mountain mama to two great kids and two great grownups
Here is a description of how I showed it to my youngest. It's important that kids have a pretty good understanding of place value first, of course. And it would also be good if they were familiar with the manipulative model for a while first. We live in Canada where we have convenient $1 coins for the 100 placevalue. You could substitute bills, or if those don't have the right feel, use buttons or "make" some dollar coins yourself. But really, in the earliest stages you could just use pennies and dimes for 1's and 10's.
I wouldn't use a manipulative that is simply labelled 1 or 10. Find a way to make a 10 (and 100) that look like ten (or 100) ones stuck together. There are various ways to do it. They sell sets of manipulatives, or you can cut out graph paper in a stick (10) or square (100), or some people use beans for ones and then ten beans stuck on a craft stick for ten. Or ten craft sticks held together with an elastic band. I used those blocks that snap together. It helps them see that a "ten" really is just ten things, not some magical transformed thing.
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I believe that one shouldn't be teaching multidigit addition and subtraction algorithms without a really firm preexisting understanding of place value. In our case we'd used manipulatives like you described (cuisenaires augmented by a placevalue rod & cube set) for explorations of placevalue for a couple of years already. I think I explained that in my blog post, but it deserves further emphasis. We'd moved beyond the more concrete manipulatives to a more abstract representation of placevalue units  but we'd done the concrete ones first, for a long time.
So your point is well taken. Kids need that placevalue understanding first. It needs to be really well understood, inside and out. Make sure your child can look at a number like 254 and know that the value of the digit 5 is fifty and the value of the digit 2 is two hundred, or that she can construct the number 8072 as 8000 + 70 + 2. After that it might be time to start talking about regrouping.
Miranda
Mountain mama to two great kids and two great grownups
Excellent point, Miranda. During our brief Montessori experience I observed some sixyearolds who were performing carrying operations (using the Montessori manipulatives of ten beads attached together) but obviously had no real understanding of what they were doing. Using manipulatives doesn't guarantee understanding of place value. Exposure helps, but I think this kind of understanding is largely developmental and is typically taught too soon.
We use the same abacus than OTMomma and it works very well with Alex (5). He can add and subtract easily. I have also printed a table with all the numbers from 1 to 99 ordered by 10. For the moment we're studying with books a the 1st grade and he can add and subtract until 10.
Isabelle HSH
If you're open to using some sort of curriculum workbook for this, I heartily recommend RightStart math. Level B would probably be right, if your child is ready for regrouping but not yet doing it. Level B does start from "the beginning" but this is important in order to grasp the RS 'way' of doing math. It's a very handson program with lots of manipulatives and games (rather than too many worksheets). And one of its greatest strengths is in how it teaches place value REALLY REALLY WELL right from the beginning (ie, the "right start").
I'll try to explain how it goes about teaching regrouping. It's fresh in my mind since my daughter just went through this part not too long ago  and she understands it PERFECTLY and loves it.
First of all, as has been mentioned above, a firm grasp of place value is a necessary prerequisite. This has been done with various kinds of representations including the abacus, place value cards, pictures, etc. One of the brilliant things RS does is use "math language" for the 10s at first (another part of the 'right start'), so instead of 'twenty' you say 'twoten', the same way we'd say 'twohundred'. So it's easy to see how if 2 apples and 3 apples is 5 apples, or 2 sticks and 3 sticks is 5 sticks, then 2 tens and 3 tens is 5 tens. And 2 hundreds and 3 hundreds is 5 hundreds. Etc. And they learn pretty early on as well that one ten is the same as ten ones; and a bit later, that ten tens is the same as one hundred. This is visually clear on the abacus  10 rows of 10 each is called 'tenten' or 'one hundred'. Once they're very clear on all this, then the 'traditional' names are introduced. But even after kids are fully using the traditional names, it still is often handy to refer back to the 'math language' names if a child is struggling or confused with a particular problem. They also learn 1000 the same way, and can see that it's the same as 10 hundreds, and also one hundred tens. Of course you can't have 10 abacuses (well you COULD but most of us don't heehee), so we use hundred tiles (a la Montessori). A stack of ten makes a thousandblock.
So anyway, your child is now confident with place value. Then there's a lesson involving cards  each card has a visual representation of the place value. One small cube, a row of ten cubes, a plate of 10x10 (hundred tile) and a 10x10x10 cube (thousand block). The kid learns how to use these to represent 4digit numbers. They are placed 2 by 2... so, say you were showing 800, that would be 4 rows of two hundred cards. This may not seem important now but it has to do with a transference that happens later. Laying out an entire 4digit number takes up quite a bit of floor space so it turns into a fun activity. :) They also do it in reverse... lay out a bunch of cards representing a 4digit number and have the child translate it to digits. Sometimes you do it where there is a zero for a certain place value  this makes it abundantly clear, where there's this 'hole', that there are 'no tens' or whatever so you need a zero there. If you're using the place value cards (which are digits that stack) this also happens automatically, so they can see the connection even if they weren't actively aware of it first. When reading the numbers, they count by 2's to see how many of each card there is, and eventually learn to just recognized the pattern of each number (in fact this is something that has already been practiced in previous lessons, numbers up to ten in dot patterns by 2's... they can tell whether a number is even or odd by sight this way as well.)
The first lesson that introduces the idea of 'regrouping' works as follows. Though we don't even use the word 'regrouping', much less 'carrying'. ;) We use 'trading!' So the problem to solve is three shepherds, who each have a specific number of sheep (each in the thousands). They're thinking of combining their sheep and want to know how many they'd have all together. How could they figure it out?
You have the numbers constructed with the digit place value cards and with the graphic representational cards (LOTS of floor space lol) then see what the child suggests. Most will figure out on their own that you can combine ALL the cards and see what you come up with.
But then there's a problem. There are something like 18 ones cards. How do you show that with the digit place value cards? My daughter was immediately gasping with wide eyes, saying "There's too many!!!" At this point, some kids will figure it out on their own, some will need some coaching. My daughter needed a few leading questions, but she did make the connection herself in the end... Remembering that ten ones is the same as one ten, you can remove ten ones from the board and replace it with one ten (from the 'bank' of extra cards)! She thought this was EXTREMELY exciting.
So after trading wherever possibly/necessary, you end up with an answer, which the child can then represent with the digit place value cards as well as write down. Voila, addition of three 4digit numbers with carrying.
This is practiced until it's secure and the child is confident with the physical trading. Then it's time to take it to the abacus, which is much faster (and takes less room lol).
For this, we use the back side of the abacus and turn it sideways. There are markings along the edge showing the place values. The 2 leftmost rows are for thousands, the 2 rightmost are ones, etc. So there are, in effect, 20 beads available for each place value. Do you see now why we did the cards in groups of two earlier? Now they can do the very same problems, but by using beads instead of cards. They can visually see groups of 10 beads  two by five  since the colour changes on the abacus. Numbers less than 10 are identified by the pattern, or else counted by twos. If there are more than 10 beads (say after an addition), then you trade  pull down the ten ones and slide up one ten with the other hand at the same time.
I should point out... at this stage, the child is doing 4digit addition with trading. But they actually have not yet learned "addition facts to 20" or whatever. At this point, they have learned how to 'split 10' (memorized 1+9, 2+8 etc) and have usually picked up a few other simple facts, but have not yet had to memorize that 4+8=12 or whatever. They are going to learn various techniques for figuring things out in their heads (for instance, 8+2 is 10, so 8+4 would be 8+2+2 or 12), but haven't yet. So the importance of understanding the concepts of place value and trading is MORE foundational, more important, than 'addition facts'. Isn't that fascinating???
Anyway, this is where my daughter is now. Give her any 4digit numbers and an abacus and she'll tell you what they add up to, and she fully understands it. We have not yet made the next step of doing it straight on paper  that will come after they've mastered how to figure out the 'addition facts', I believe. But you can hopefully see how it will be a simple transference. They have been writing down the addends and the answer all this time, only the calculation is done on the abacus. Doing the calculation in your head instead is just the next step.
Shall I also mention that my daughter is just 4 years old? She is "mathy" for her age, for sure... level B is generally considered a 'grade 1' program. But she's still not turning 5 until next month and she fully understands adding 4digit numbers and trading. I *love* this math program!!! And so does she  we unschool in the sense that I don't expect any particular schedule or whatever from her, but she will ASK to do a math lesson most days! :) And if you're NOT looking for a curriculum at this point in time, hopefully my explanations of how they explain things will have given you some ideas?
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