Daily heroic on 26 or 28, public events, and vault of glass are the primary means.
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VoG is something you need to do with teammates it seems to me and I don't have any friends with an Xbox One that play Destiny
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That is correct it takes a team. My suggestion is public events. The first event you do each day gives a chance for a shard at the postmaster reward. Each completed event also gives a small chance of earning materials as soon as you complete it.