Turning left on Monument in the Fan - what's the correct way?

I see some people stopping in the median, not for oncoming traffic, but for what I'm assuming is for the other light to turn green.

So, say they're headed toward VCU from the West End. The Monument light turns green, they pull into the median and stop, waiting for the next light - the cross street light - to turn green.

Is that a thing? Am I clueless? I figure it's like any other left turn and you just proceed through when there's no oncoming traffic.