I hear this question so often! This "new math" isn't how I learned it, and it probably isn't how you learned it either. You know what? That's totally ok.
This new math is basically relating math to place value. It is so important because we, as a generation, are pretty awful at it. So our kids will begin to understand the WHY behind why we do things like regroup when subtracting or why we bring a number down during division.
In my classroom, I strive to make sure the students see the real world connections to our math. Let's be honest, we can't survive without it.
So, new math is just showing the relationship between each problem and place value. For example, if you were subtracting 56-43 the old way, you would subtract 6-3 and then 5-4, right? The problem with that is that you aren't taking 4 from 5, you are taking 40 from 50. So when we look at that problem in class, we do 6-3 and 50-40. Then answer is the same, the process is a little different.
I want my students to see a number of ways to do math (especially in multiplication and division). In fact, we don't teach the standard algorithm for multiplication until 5th grade. So we will see how to move from what they've learned so far, to what I learned well over 20 years ago.
Don't panic when you see this math, and please don't feel like you have to show them how you learned. Yet. In fact, let them show you how they are learning and ask them the magic question...WHY?