Track - 6 weeks later

Six weeks ago I attended my first track workout.  As a complete beginner to track I had to learn what 200's and 400's were, and how to avoid starting out too fast and fizzling before the end of the session.  I was also given modified assignments since I was not only new to track but also not in peak running condition.
The word on the street is that regularly incorporating track sessions into your weekly routine can make you a faster runner.  Coach said to give it a month before I should start expecting any improvement.  But like everything Coach says, I have to translate that into my very specific Jill-Dog Years-Fat Ass-multiplied by 3.14 slices of pie- formula.  I waited six weeks to reflect.
So here we go - the six week results:
Week 1:
200: untimed
200: untimed
400: 1:46
400: 1:46
400: 1:43
At the end of that first workout, I was definitely depleted.  I couldn't have run another 400 in under two minutes if Channing Tatum was waiting for me at the finish line in nothing but a bow tie.
In weeks 2 through 4, we added different distances and increased the number of intervals, but I still never completed an entire workout.  Week 5, I finally completed the entire workout, did my first 1200, and managed to still match or beat my previous weeks' times.  Holy smokes!  And that brings us to...
Week 6:
200: 47
400: 98
800: 3:38
800: 3:38
800: 3:36
400: 98
200: 43
I'm pretty psyched that I managed to shave significant time off the 400's (and the 800's, which I did in 3:51 on week 2).  I'm also pleased that my times stayed consistent or improved throughout the workout, I completed the entire workout, and I didn't even vomit doing it!  For me, an exciting milestone is getting my 400 times into double digits.  Coach counts anything under 1:40 in seconds, so I slid in right under the wire.  Go legs!
About halfway through last night's workout a family with some kids on scooters and with soccer balls showed up at the track.  I figured they would get in our way but they were pleasant, and I overheard the kids say, "wow they run so fast!!"  After we were done a boy of about 7 or 8 came up to us asking if he could do a lap on the track.  The coach timed him while we all watched and waited for him to come back around.  We clapped for him, and would you believe he did it in 89 seconds!  That little squirt blew my doors off!  You could tell it totally made the kid's night to be running with the "track runners", and he asked if he could run with us again next week.  I think we have a new recruit!


So, in my uneducated opinion with nothing but 6 weeks of modest experience to back me up, if one were to ask me how to run faster, here's what I would say:  Running faster makes you faster.  I think if you run 15 miles a week at a 9:00 pace, you probably won't finish a 5k at a 7:30 pace.  You might, but it will probably hurt a lot.  Again, I'm speculating.  I think if you want to run faster you have to train your body on how to adapt to faster paces and sustain it, which can be done through these grueling intervals that just keep getting longer and longer.  Again, I'm just thinking out loud.  Or, maybe all it really takes is the heart of a 7 year old.