BY FRANCES ROBLES

 WASHINGTON -- The impromptu school set up for Elian Gonzalez and his
 Cardenas classmates at the Wye Plantation is the newest cause for contention
 between the U.S. State Department and Cuban diplomats, who were told
 Thursday that the boy's study sessions are ``counterproductive and reckless.''

 ``It's sticking a thumb in our eye,'' one State Department official said.

 Each morning at 8:30, Elian dons his Cuban school uniform and spends mornings
 learning math, gymnastics and other subjects. He's joined by four friends and
 taught by a familiar face: his kindergarten teacher from last year in Cuba.

 Cuban diplomats were called to the State Department Thursday and told that the
 school setting was inappropriate. It followed an earlier warning that Elian's
 teacher, Agueda Fleitas, should not educate children unless she has a valid
 teaching certificate -- from this country, State Department sources said.

 Publicly, at least, the State Department is at odds in the Gonzalez case with
 other government agencies, whose officials have declared the goings-on at the
 Wye Plantation none of its business. In fact, the Cubans responded to the State
 Department by brandishing newspaper articles quoting an Immigration and
 Naturalization Service spokeswoman dismissing the issue.

 ``How Elian is educated is up to his dad,'' INS spokeswoman Maria Cardona said.

 Officials say the INS and Justice Department do fear that the existence of the
 school could hurt their case before the 11th U.S. Circuit Court of Appeals, but
 they are unwilling to take any action because they do not believe the school is
 illegal.

 That position dismays the State Department. ``So, what, anybody can set up a
 school anywhere?'' a State Department official said. ``The state of Maryland has
 not thought this through.''

 How Elian should be schooled is yet another issue raised by those who oppose
 his father's desire to return him to Cuba and are critical of the way the case is
 being handled. Elian's Miami relatives and their lawyers have complained bitterly
 about the so-called indoctrination they believe Fleitas is performing at the Wye
 Plantation.

 `INDOCTRINATION'

 The family's attorneys filed an addendum to the court record showing photos of
 Elian at Wye in his Cuban school uniform, with the blue scarf of the Pioneers
 youth organization.

 ``Anybody would be better than the indoctrination he's getting now,'' attorney Jose
 Garcia-Pedrosa said of the teacher. ``Whether that's in the form of public schools
 of Maryland I don't know. Our main concern is the brainwashing.''

 Cuban Interests Section spokesman Luis Fernandez said he knew nothing about
 the matter. Juan Miguel Gonzalez's attorney, Greg Craig, did not return telephone
 messages.

 Thursday's edition of the Communist Party newspaper Granma said the school
 was arranged to keep Elian from falling behind in his studies and to free him from
 the Cuban exile community's grip.

 It's ``to save Elian not only from the clutches of the empire and the Miami Mafia
 but also from the clutches of ignorance, lack of culture and selfishness,'' the
 paper said. ``The teacher went there, with her books and other educational
 materials, to keep the boy -- who had been detained in Miami for five months --
 from missing any more classes and falling behind in his education.''

 TUTORING CLEARED

 The INS, the U.S. Department of Education and the Maryland Department of
 Education gave clearance to Elian's private tutoring. They argue that he is not an
 official resident of the United States subject to state laws which would require him
 to attend public school.

 ``This is not something Queen Anne County or the state has any ability to
 enforce. The question is, where is he a resident of?'' said Neil Greenberger, a
 communications specialist for the Maryland Department of Education. ``It's not
 something the state is compelled to answer in May, with two weeks left in the
 school year. School starts the last week of August. The state of Maryland would
 hope Elian Gonzalez is somewhere else by then.''

 But if a lengthy court battle keeps Elian here for months, would Juan Miguel
 Gonzalez be bound to check his son in with the Queen Anne County Board of
 Education? Nobody seems to know.

 Home schooling is perfectly legal, but state law says a parent or guardian must
 sign a contract showing that the child is being taught curriculum appropriate for
 his grade. Nobody from the Wye Plantation or the federal government has
 contacted the local school board in regard to Elian's classes, Greenberger said.

 WAIT AND SEE

 Had Elian enrolled in the 7,000-student district, he would have been assigned to
 Centreville Elementary School, a half-hour bus ride away where the principal
 already has a Spanish-speaking playmate in mind for him.

 ``Let him get a little bit of peace,'' groused Dominic Romano, the district's
 supervisor of pupil services, who said he had no interest in taking on the issue.
 ``He's been subjected to too much already. Let's see what happens with the
 courts.''

 INS spokeswoman Cardona said Elian's instructor does not need Maryland
 teaching certification.

 ``If she were purporting to teach American children, or non-Cuban children, of
 course it would be different,'' Cardona said.

 Maryland education officials say they may view the situation differently if the boy
 is still here in the fall, when he would be scheduled to start second grade.

 ``It's not as simple as moving your car from state to state, where they tell you that
 within 60 days, you better get new plates,'' Greenberger said. ``That law is clear.
 This is not clear. We can't just put a license plate on Elian Gonzalez.''