"glean some insights" is correct and usable in written English.
You can use it when you are looking to learn something new or gain a deeper understanding of something. For example, "I took the time to read through the report thoroughly in order to glean some insights into the economic trends of the region."
But Payton could glean some insights from Belichick's tactics.
That hasn't stopped every other Tom, Dick or Harry from persistently questioning the company's decisions post-Jobs, trying to glean some insights from his management style to apply to their own business or attempting to cash in on his name.
To glean some insights into the roles of the sites that positive selection and functional divergence might have, we mapped these sites onto the model as well as along the sequence alignment (Figure 3, B, C and D).
I tried to glean some insight from my discussions with women who are personal finance and parenting experts.
Do you want your readers to finish your book in one sitting or work hard at every sentence to glean some insight from it?
They flock to Robert Johnson's crossroads, to Graceland, to the Chelsea Hotel, hoping to glean some insight into the music that moves them.
I spoke to a few very successful people working their own "dream jobs" hoping to glean some insight.
But could we glean some insight into ourselves in this process, some greater understanding of the connection between body and mind?
Interestingly, as we look back at Brown's books prior to Daring Greatly, we can glean some insight into how she became such a champion of vulnerability.
But you probably figured you could visit her website to glean some insight into her vision for California (after all, she IS running for governor of California, right?).
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