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My teacher told me to rotate my arm and wrist and stretch my thumb under which is the complete opposite to the advice on the piano-ology video.
I'd be interested to know what you think
The method you described is the standard way of doing it and eradicates the need for any use of the sustain pedal.
He certainly spent a lot of time talking about this 'pianological philosophy' but he's thinking way too much about it, not that it has done much good. There are some good ideas about using fingers to pivot and rotation of the wrist for example, but this should be viewed with caution imo.
OK so if your upper arms are at about 45 degrees to the sides of your body, the slope from your shoulder to your elbow is also about 45 degrees, and you are relaxed, you should now be able to take that 2nd finger that is currently playing the G, and play the E above the C. That is the hardest part of the arpeggio.
Now, you may have to modify what you do slightly because you are probably a different size and shape to me.
Legato isn't just about connecting the notes, which you can obviously do with the pedal, it's about the sound relationships between the notes, and thats about the ear and the fingers.
The piano-ology website doesn't actually teach you to play a legato arpeggio. It teaches you to play broken chords slurred in three note groups, using the pedal to fill the gap. Different sound.