Feldenkrais taught group Awareness Through Movement sessions in his studio on Alexander Yanai street in Tel Aviv, Israel starting in the mid 1950's. Many of the sessions were recorded on reel-to-reel tapes, which was cutting-edge technology at the time. Feldenkrais recorded and edited the sessions until he got them exactly how he wanted them. And his students would do the sessions via the tapes in the studio.
After Moshe's death the tapes were transcribed into written Hebrew, the language in which he taught in Israel. And later the transcriptions were translated into English and published as the "Alexander Yanai" sessions. When you hear people in the various Feldenkrais communities talking about the "AY" or "Alexander Yanai" sessions they are talking about sessions based from those tapes and transcripts.
I (Ryan Nagy) record and share many sessions based on the AY sessions on my Easy Feldenkrais site. To me they represent a very fertile time in Moshe's life when he was experimenting with how to teach his work. You could consider the sessions experiments where Moshe was putting his ideas into practice and seeing how people responded to them.
An interesting idea to consider about the Alexander Yanai sessions is that Feldenkrais rarely corrected people. He would most often let students make whatever sense they could of his instructions and he would simply watch. Though..you can see from reading the Alexander Yanai transcripts that Moshe would occasionally get frustrated with what he was seeing and called people out, "Oy Vey!."
Very First Draft, December 26th, 2017. Updated January, 12th, 2018. Much more to come.