My kid listens to his teachers, but won't listen to me.
All year I listened with glee while my six-year son's kindergarten teacher and after-school instructor told me how wonderful and attentive he was during class. They described him as bright, responsible, excellent in math, ready to help others and always making sure the classroom was clean. What child are they talking about? Every day this summer, I give him a little bit of homework to do to keep his brain active and prepare him for the 1st grade. He goofs off. He argues. He acts like no one gave him any instruction for the past 9 months and he flat-out tells me I am not his teacher. When I told a few days ago that I am responsible for his safety (he was angry I did not let him run across the street with his friends), he said "you're not one of the summer camp volunteers." WTF? Did we actually have twins? Is there a kid who goes home with me and another who goes somewhere else? What do i do?
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My 10 year old grandson appears to be the model student.
Very bright, makes excellent grades, never any disciplinary problems...in school.
At home, he will argue with his mother about some trivial thing until he is blue in the face.
Not so much with his dad, but sometimes.
As a 2-5 year old, he spent more time in the 'tine-out chair' than he did walking around.
Kid was in stir half his life.
When I was a kid, 60 years ago, teachers were merely an extension of your parents, who were the ultimate authority and say-so.
I don't know when or why it changed.
Lack of corporal punishment?
Based on past experience, I knew that if I crossed a certain line I'd get a sound swatting on my bottom.
I know this is no longer acceptable in most circles.
It's true that boys are a harder nut than girls, generally.
Our daughter was pretty much a joy to be around.
We'd tried to keep 'rules' to a minimum so we didn't stifle her.
She was usually very agreeable to the ones we laid down, especially when we explained why.
I wish I knew what the answer was.
My daughter is having a tough time right now.
I have had parents tell me that their child is an angel at home and only misbehaves in my class! 
Sounds like he is feeling his oats and after being so good for 9 months wants to let loose a little. Maybe you could fore-go the summer "homework" in favor of more reading, or more practical math, like shopping, money, etc. Give the kid a break! LOL
the younger one, who has ADHD.
As to what to do, I have yet to solve it. *sigh*
As far as "homework," hubby and I are both trained teachers, though neither of us are teaching now. I have the kids watch "learning shows" on Discovery, etc and play learning games on the computer. There are many available which tie into favorite TV shows like "Sesame Street" and "Bob the Builder." There is also a series called "JumpStart" which our kids loved. Another thing we have done is play "Bakery" using playdough, or "Store" using everything in the house. Nature walk scavenger hunts were also a lot of fun! (It's really important that you are participating in all of these activities too. This counts not just as learning time but quality parent/kid time.) Making up your own stories and illustrating them or trying MadLibs can be great. Just translate noun into "thing" or adjective into "describing word" and give some examples. They are very concrete thinkers at that age. You may just have to ask him to point to something in the room and describe it, and use those words. (Also, fill in several blanks yourself to keep from frustrating him and add variety to the story.)
"I spy" and "I packed my grandmother's trunk" are fun games, although at that age you can't expect a child to remember the whole list. (We played last night and at our age couldn't remember the whole list!) "Hot and Cold" is always a great game. I used to hide a Hot Wheels car and guide them in searching for it. You can also find website for age appropriate science experiments. Anything to do with rockets and balloons went over well here, as did the cyclone in a bottle experiment. They also started collecting rocks at a very young age, although it was later when they started learning to identify them. You can fill a container with sand or rice, however, and hide pennies or rocks in there for him to find. He has to count the pile before and after to see if he found everything. Usually 20 was my limit for items to hide.
Limbo/jump the rope or stick is great, and so is "Statue" where he moves around to the music and when it stops he has to freeze. Playing toss is great! Try to set a high score and count every catch! Build a city with blocks, if you have them.
(I am sure there are more activities we used to do that have slipped my mind. I had a great time walking down memory lane, though! Thank you! 
)
that is odd! 
Go into stasis until he reaches 21 or so. Even then he may not listen to you.
