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I am personally not a fan of Trachtenberg. For some reason it doesn’t click with my way of thinking.
That does not mean it is bad though. Try it out and see what works best.
This forum is filled with hundreds of answers for your questions.
The best advice I can give you is try everything and see what clicks with your mind.
Some like Arthur Benjamins system (google it or ask me) for multiplication, some like the criss-cross method better.
Try both and see what you like.
Also let me know if you want to do everything in your head without seeing the numbers or that you see the numbers on papers when calculating.
There is a difference in the sense that if you do everything in your head without seeing the numbers on paper you need to keep more information in your head and this means you need to start with simpler calculations. Don’t do 4 digit multiplications then. Start with 2x2 and then see if this comes easy.
Ask a lot of questions.
We can help you with everything.