I love math. I always have. I think the reason I love math is that all of its rules seem very straightforward to me. It always surprised me when I work with a child who hates math or doesn't get it. What I have realized is these children didn't understand something that is necessary to be able to make sense of the math they are currently trying to do.
So, if you have a child that is having difficulty carrying and borrowing, I would guess you have a child who cannot add and subtract around 10 quickly and easily. In my opinion, this is the pre-requisite to being good at carrying and borrowing that is the obstacle to feeling that carrying and borrowing are easy for many children.
What is adding and subtracting around 10? Here are some examples:
Adding examples:
8 + 5 = 13
7 + 9 = 16
This is the adding skill to practice. When your child is able to add two single digit numbers whose sum is greater than or equal to 10 quickly and accurately, they will be able to learn carrying easily.
Subtracting examples:
16 - 9 = 7
13 - 6 = 7
Subtracting around 10 starts with a number between 10 and 18 and subtracting a single digit number so that the answer is less than 10. When you child is able to subtract around 10 easily and accurately, they will be able to learn borrowing easily.
There are many ways to learn these. If a child is supposed to be ready to carry and borrow, but don't know these math facts, the method that was used at school, or by you isn't working for them. Some children like flashcards, others hate them. Some children like to fill in worksheets, others are willing to practice orally with you in the car. Other children like it to be turned into a game. I have had good success with elementary children playing Concentration type games where the child turns overs the numbers and the answers to win, or bingo type games where I would call out, "16 - 7" and then they would need to find an 9 on their bingo card.
Once, your child can do this, the next step is to practice carrying by adding a digit to the first number. For example, 38 + 7 - 45. When this seems easy, the next step is to make the second number 2 digits as well. For example, 46 + 29 = 75. You child will be successfully carrying. The only other complication is to add more digits, but if they get the concept with two digits, the difficulty with multiple digits is attention to details and focus.
Borrowing follows the same progression. Increase the tens digit with the first number. For example, 37 - 9 = 25. The next step is to make the second number two digits as well. 54 - 38 = 16. Then it's on to bigger numbers with more digits.
The key to overcoming math difficulties is to figure out the underlying math skills that the child is missing that is making the task at hand difficult. Once the underlying difficulty is solved, the child should be able to be successful with the math task at hand.
Good luck!