Almost everywhere you turn in this state, schools are in trouble. In Toledo, they're scheduled to shut down next week - perhaps until January. Here in the state capital, where to reopen next fall signs on buses ask, "Do you still care about schools?"

And in Cleveland, where schools are flat broke officials today were faced with a no-own decision, whether to close down for the year and defy a federal judge, or to defy state law by staying open and being personally liable for some $400,000 a day in expenses.

"We're caught between a rock and a hard place." Arnold R. Pinkey, chairman of the Cleveland Board of Education, said today. "I don't know what we're going to do. We're going to have to talk with our lawyers."

The problem everywhere is money. A record high 46 Ohio school district with over 312,000 students are facing serious financial troubles this fall and have asked the state auditor to review their books; 27 have been given permission to close. In addition, voters in 22 school districts will be asked to increase their property tax rates next month. If past experience is a guide, at least half of the rises will be rejected, compounding the problem for next year.

Nowhere is the situation more critical than in Toledo and Cleveland, two of the state's largest school systems with enrolments respectively of 54,000 and 110,000.

Ironically, the financial plight of the two systems is not unlike that of dozens of other big-city school systems across the country. They present a problem of increasing national concern and no easy solution.

"The health of large city school finances is precarious," declared a recent report by the National Conference of State Legislatures. "City tax bases are stagnating or declinig. Minority peoples, many of which require high-cost programs, remain a large and growing proportion of city school enrollments."

Two quirks in Ohio law, however, make the situation here far more dramatic than elsewhere. One forbids any district to enter a new calendar year with a district. The other requires that local taxpayers must approve all property tax increases.

Toledo schools are to shut down next Friday when they run out of money. If voters approve a 6.1 mil property tax increases, they'll reopen Nov. 9. Otherwise, they'll remain closed till January.

Cleveland schools went into the red this week, according to officials, and were to close this Friday. They mounted an intense lobbying effort here this week to apply what was called a "Band-aid solution" to their financial cancer.

But the State Senate here yesterday scuttled the move, which would have allowed Cleveland and other embattled school systems to borrow their way out of their 1977 depts. This occurred just hours after a federal judge ruled that Cleveland schools must remain open even if the Board of Education there says it's tun out money.

U.S. District Court Judge Frank J. Battisti, who is overseeing a Cleveland desegration suit, declared that school officials had been imcompetent in handling public money, and called the closing an "indefensible position." he ordered a independent oudit of schools finances.

The judge also said school officials "have victimized virtually everyone involved or concerned with Cleveland public schools" and had squandered money "on maintaining segregation and defending it in this court."

His ruling stunned school officials. In lobbying here, they had maintained that their central problem was "cash flow" and if they could be given emergency permission to borrow money until next year they could balance their books - and keep their schools open.

If they were to do so without permission from the legislature, their lawyers have advised them, they would volate state law and could be held personally liable - to the tune of $400,000 a day. If the board obeys the court order, finance chairman George Dobrea said, it will be unable to repay $15 million due by the end of the year, and thus will also endanger its credit rating. "The real question is how the decision affects the ability of school systems in Ohio, and perhaps the nation, to borrow money," he said.

The murky situation remained unresolved tonight without Cleveland's 110,000 school children knowing whether or not Friday would be their last day of school this year. But the school board tentatively scheduled a meeting indicated in interviews that schools will probably remain open - at least temporarily.

A citizen's committee, set up to study the plight of Toledo schools, addressed itself to this harsh reality earlier this month: "The system is near bankruptcy and simply cannot continue its present level of services."

The problem has been building for years, compounded by the future of voters to approve a single property tax increase since 1968, Toledo Superintendent Frank Dick said this week. "What's happened is the public [didn't want to believe anything like this closing schools] would happen. People thought some miracle would save us. A lot of them still do."

Dick, superintendent for the last 13 years, fearing his personality had become an issue, resigned recently to "clear the air," in hopes that this would help chances of passage of the tax increase.

But the measure remains in real trouble. "If the vote was held today, ther's no way in hell's name it would pass." Dal Lawrence, president of the Toledo Federation of Teachers, said yesterday. "People are dissatisfied with the superintendent. They're dissatisfied with teachers' salaries. They're upset with too many buildings being built, and too many minority students being bused."

Meanwhile, morale at places like Waite High in a blue-collar neigborhood of eastern Toledo is low. Teachers complained their salaries haven't increased in two years, and that a long-term closing "means the death of public education" in Toledo. Students complain they don't want to be thrown out to school in mid-year, disrupting their studies and extracurricular activities.

"I don't think anybody cares about kids anymore," Stephanie Villa, a 15-year-old junior, said today. "It seems everyone you talk to is against it [the tax levy] . . . it's garbage. I'm losing an education over this."

Whether this attitude will change after schools close next Friday is a matter of great conjecture in Toledo. Superintendent Dick maintains it will. "Voters are frustrated. Many of them are older people living on fixed incomes. They've seen inflation come in." he said. "They're distrusful of government. But I think when the chips are down they'll support our schools."