Lawn burweed removal

Hi guys, was wondering if anyone knew how to get rid of lawn burweed. I live in north Mississippi and this stuff showed up in my lawn about 3 years ago and has spread greatly every year. I'm not picky on what is growing in my lawn and would leave it but the stickers on these things are unbelievable! With a 2 yr old running around and helping me in the raised bed, I can't even let her enjoy going barefoot. Pretty much everything I'm finding on this stuff is recommending chemicals to kill it and I'm not going that route. Any ideas on how to get rid of it?
Here is a link to the plant I'm referring to.
http://www.ipm.ucdavis.edu/PMG/WEEDS/lawn_burweed.html

Howdy David, welcome to permies! Paul has a lawn care article at www.richsoil.com that may shed some light .
I guess the way I would aproach it is this. What happened 3 or 4 years ago that would give this weed a chance to invade? Did you open up a space that mother nature filled with this plant?
Now that it is growing what can you do to outcompete it and send it back into dormancy. Can you plant something that will grow taller or thicker? It says it regenerates from seeds. So can you stop the seeds from germinating somehow? Does any of that make sense?

Thanks for the link miles, I'll check out that write-up. As far as something that could have happened.... I did start cutting my grass on a lower setting a few years back and I really think that it could have been the problem that opened up this whole issue. As far as growing something taller, I did raise my mower height and am going to try not to cut as much this year. I read somewhere about vinegar, but it supposedly kills everything from what I understand so that's why I decided to see if you guys had any ideas I had missed.

There you go. Let me know if leaving the grass taller makes a difference.

I will update and hopefully it will work! Thanks