The "Old" Math vs. The "New" Math

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I've always heard about the "New" Math but I never really knew what it was until I listened to this song by based kike Tom Lehrer:
https://www.youtube.com/watch?v=UIKGV2cTgqA
He makes a big thing about the "New" Math method being complicated but it turns out this is the method I always knew.
When he's talking about the old (to him and his audience "normal") method none of it makes sense and his joke about class really just flew over my head.
Obviously the new math method is much more logical (really they are both logical otherwise one method wouldn't mathematically work but you know what I mean) which is why it was adopted but was they old way quicker for mental math and stuff?
How extensive was the "New" math's reforms?
Are there any old math-fags here that know how we used to do things?