At the heart of the teachers' strike that has disrupted the District school system for the last two days lies a long-simmering conflict between the Washington Teachers Union and a school board intent on regaining full control of the school system from the teachers who work for it.

A common belief is held by a majority of the board and union members: the city's public schools are doing a poor job of educating the District's children.

But the board and the union have widely different ideas about what can be done to improve schools and stop endless tales of D.C. public school graduates who cannot read bus signs, pass minimal literacy tests to get into the Army or speak and reason well enough to keep a job.

Those differences boil down to board members' feeling that they must have full control of what goes on in classrooms if they are going to improve the school system and teachers' feelings that since they are in the classroom dealing with students, they should be deciding how the schools are run.

The union has acknowledged that it is pleased with the current contract, which reflects gains made in benegits and influence since the union began collective bargaining with the school board in 1967. But the board has said it is dissatisfied with the contract, which expired in January 1978, and with concessions made to the union in past contracts with other boards.

When the union asked the board in February for a fourth extension of the contract while negotiations continued, the board refused. After ignoring a strike threat by the union and refusing to give the union the extension it requested, the board told School Superintendent Vincent E. Reed to cut off automatic union deductions on teachers' paychecks and end all provisions of the expired teachers' contract.

The union, consistently threatening to strike if the board ended its automatic dues checkoff, saw the board's move as a provocation, a dare to strike from a get-tough school board.

And throughout the union's strike threats the board held tight at the bargaining tabel to its resolve to lessen techers' influence on the school system.

Now there is a teachers' strike and the question is whether a hard-line school board decided to force a strike with the union.

"We didn't try to force a strike," said Alaire Rieffel, school board member from Ward 2. "Under board rules, we can only negotiate working conditions for teachers. It was illegal to put in those articles (in the old contract) on students' discipline and grades in the first place. They should never have been in there and this board is going to get them out. If that is taking contro, then I guess we are taking control."

"The board has got to stop this foolishness about putting teachers in their place and teaching us a lesson," William Simons, president of the union, said at a union rally last Wednesday night before the strike began. "Negotiations are not the place for putting anybody in their places."

At another union meeting, Simons, a 55-year-old former social studies teacher who has been president of the union since 1964, said the board is not regarding teachers as "professionals who should control our working conditions, we're the ones in the classrooms..."

"The board wants to take away our right to grade the students in our own classroom," Simons said, his eyes wide with amazement as he spoke to union members. "They want to take away our right to discipline students in our own classroom."

The board has contended in negotiations that school discipline should be the concern of students and their parents as well as techers and school administrators, and has proposed that new disciplinary procedures be determined in discussions involving members of all four groups.

The union and the board have been in opposing stances since the last contract expired 14 months ago. As the contract was extended, both sides were aware that the other had a very different idea of what should be in a new contract. In October, even before negotiations had begun, the teachers' union almost called a strike in a preliminary skirmish with the board over the ground rules for the talks.

Since the talks began, neither side has shifted its stance.

A majority of the school board still believes that to improve the schools -- and end criticism of the board by politicians and parents -- the board must have full authority to make decisions about how schools are run and must be able to demand hard work, and accountability for that work, from all their employes, particularly teachers.

Most board members think that the teachers union, through collective bargaining that began in 1967, has taken away much of the board's authority and given it to union committees and teachers at individual schools. Board members said they negotiated away much of their authority in contract talks with teachers because they do not control teachers' wages and in the early years of bargaining did not have professional negotiators to bargain with the union.

"The board is reponsible for the education of children in this city," Reed, the superintendent, said yeaterday.

"And when the educational system fails, the only people held responsible are the board members and the superintendent.... The board wants to have as much clout as possible to fulfil its responsibility without any interference."

The union, in turn, says that teachers, who are on the front line in efforts to educate children, should be consulted about any decisions made by the school board or school administrators about how schools will be operated.

"The teachers are in the classroom," Simmons said in an interview last fall. "The members of the school board are downtown. Many of them haven't been in a school in years, and yet they are saying that we should shut up and let them tell us what should be done in schools....

"Their administrators want to tell teachers what to do. They want teachers to fill out this form and that form and they want us to stand in the halls.... Some of the principals won't give teachers their paychecks when they arrive. They say you have to wait until the end of the day. Those little dictators treat teachers like children and then they blame the problems of the whole school system on teachers. It is that ludicrous."

Since the union first won the right to represent teachers in collective bargaining with the school board in 1967, Simons has fought to increase the amount of influence teachers have in decisions made by school administrators and the school board.

The unions' first attempt to influence what was being done in schools was a plan called More Effective Schools, which attempted to set class size for schools and other policy matters.

From the union's inception, Simon's saw it as a dissident voice. After his union defeated the National Education Association to be the bargaining agent for teachers in the city, Simons told reporters: "I think this vote is clearly a mandate for change. It is a protest against the way the public school system in Washington is now being run."

In 1970, union opposition was the prime cause for the failure of the "Clark Plan," a program designed to bring students in the District up to grade level test scores in reading and math. The plan included a proposal to have teachers' wages set on the basis of "merit," with teachers who succeeded in raising their students' test scores rewarded with higher pay for their accomplishment.

Since 1970 the union has continued to make gains in its contractual benefits and in wages for teachers. Despite a 1971 agreement to accept more work if they received a pay raise, the union has not agreed to work longer hours or a longer school year. But since that time they have had wage increases.