I believe the story you are after is the short story The split cherry tree by Jesse Stuart written in 1939.
I found this by searching 'story boy father school teacher microscope' and it was the fourth or fifth hit.
I read the story and as I couldn't find a good conclusive summary online I wrote one myself:
The story is set in rural America and is about a boy named Dave Sexton who climbs a neighbors' cherry tree with a group of his friends. The tree breaks and the boys must pay for it. Dave Sexton can't afford to pay the dollar fine, but his schoolmaster, Professor Herbert, offers to pay the fine if Dave stays two hours after school. Dave however believes he has to do chores on his father's farm, and that he would be whipped if he didn't do them. Proffessor Herbert says he is too old to be punished like that.
When Dave gets home, his father, Luster, is furious with Dave and professor Herbert, and with schools in general. He sets of to the school with a gun saying that a bullet will make a hole in a schoolteacher as well as any other.
However Professor Herbert manages to get Luster to look through a microscope at something he scraped off his teeth. Luster is amazed at what he sees and his view on schools changes entirely.
The story ends with them both going home and Luster telling his wife all about what he saw.
The full story can be read here.
Here are a few relevant extracts:
Pa stopped scattering the bundles of fodder. He looked at me. He says, "Why are you gettin' in here this time o' night? You tell me or I'll take a hickory withe to you right here on th' spot!"
I says, "I had to stay after school." I couldn't lie to Pa. He'd go to school and find out why I had to stay. If I lied to him it would be too bad for me.
"Why did you haf to stay atter school?" says Pa. I says, "Our biology class went on a field trip today. Six of us boys broke down a cherry tree. We had to give a dollar apiece to pay for the tree. I didn't have the dollar. Professor Herbert is making me work out my dollar. He gives me twenty-five cents an hour. I had to stay in this afternoon. I'll have to stay in tomorrow afternoon!"
"Now, Dave," says Pa, "let's get ready fer school. I aim to go with you this mornin' and look into bug larnin', frog larnin', lizard and snake larnin', and breakin' down cherry trees! I don't like no sicha foolish way o' larnin' myself!"
Pa hadn't forgot. I'd have to take him to school with me.
In the afternoon when we went to biology Pa was in the class. He was sitting on one of the high stools beside the microscope. We went ahead with our work just as if Pa wasn't in the class. I saw- Pa take his knife and scrape tartar from one of his teeth. Professor Herbert put it on the lens and adjusted the microscope for Pa. He adjusted it and worked awhile. Then he says: "Now Luster, look! Put your eye right down to the light. Squint the other eye!"
Pa put his head down and did as Professor Herbert said. "I see 'im," says Pa. 'Who'd a ever thought that? Right on a body's teeth! Right in a body's mouth. You're right certain they ain't no fake to this, Professor Herbert?"
"No," says Pa. "You are goin' to do as he says. He's a good man. School has changed from my day and time. I'm a dead leaf, Dave. I'm behind. I don't belong here. If he'll let me I'll get a broom and we'll both sweep one hour. That pays your debt. I'll hep you pay it. I'll ast 'im and see if he won't let me hep you."
I looked online for illustrations but couldn't find any...
