The dog days of summer have been a relentless host to the new school year in Maryland's public schools. Unseasonably hot weather forced hundreds of thousands of public school students to stay home or cut short their school days last week, marking one of the steamiest school beginnings that education officials can remember.

But even before students returned this fall to their sweltering classrooms, which reached 100 degrees in some schools in the Washington suburbs, heat of a different type was taking its toll in other quarters.

Over the summer, a new superintendent in Baltimore City had to respond to criticisms about the city's handling of education for the handicapped and teacher complaints about classroom sizes.

Montgomery County's new schools chief, only a few weeks after taking office on July 1, had to extinguish the first flames of a racial integration controversy. In Prince George's County, the school year opened with the expiration of the teachers' contract and an informational picket by the teachers' union. There is talk there, although not terribly serious, of a teachers' strike.

As if that were not enough, Anne Arundel County joined the fray last week.

Faced with no cost-of-living increases in their salaries this year, Anne Arundel teachers and other school employes are demanding additional benefits and have threatened to ask the Maryland Board of Education to intervene if a settlement cannot be reached with county officials.

While the summer heat has symbolized tensions on the education front across the state, election year politics have drawn more than the usual attention to the schools in Baltimore.

Education was a key issue in Tuesday's citywide elections. During the last few weeks of the electoral campaign, Democratic mayoral challenger William Murphy accused incumbent William Donald Schaefer of promoting downtown businesss development at the expense of the neighborhoods and the predominantly black public schools. The Baltimore Teachers' Union also got into the election hoopla, petitioning unsuccessfully to get a ballot measure to reduce class sizes.

The real dog days for Baltimore's school system predate the mayoral campaign, however. The school board horribly botched a year-long nationwide search to replace Superintendent John L. Crew and finally turned to an insider, Alice Pinderhughes, to take the unenviable position.

Pinderhughes, a 40-year veteran of the system, an expert in child education and the widow of a former assistant superintendent, was named May 21. She became the fourth black woman to head a major city school system (the others are Floretta McKenzie in the District, Constance Clayton in Philadephia and Ruth Love, who is paid $120,000 annually as chief of the Chicago schools). Backed by Schaefer and highly regarded, Pinderhughes, nonetheless, will have to prove how tough she can be with City Hall.

She had her first test over the summer, when advocates for the handicapped launched a full-scale attack against the school system, saying it has failed to provide adequate services to the handicapped. Threatened with a lawsuit, the city (with help from the state) coughed up some money and after a few weeks of tumult, all parties appeared satisfied.

In Montgomery, Superintendent Wilmer S. Cody already has won kudos, even from some early skeptics, primarily for his handling of what is known there as "the Larchmont controversy."

In the thick of the fight between community factions, the board and the County Council, he was a moderating influence. He showed a style markedly different from that of his predecessor, Edward Andrews, who masked terrific political savvy and bureaucratic manipulation behind a "good-ol'-boy" exterior.

Cody's style is more gentle and understated--he certainly could not be described the way Andrews once was: "dumb like a fox"--and he has given board watchers the distinct impression he is more intellectual and analytical than Andrews.

The Larchmont issue, which he was credited with solving in a few weeks, centered on whether the county's plans to lease a closed school building to a private school would jeopardize integration efforts in the area. Cody's successful maiden voyage into this issue was no mean feat in Montgomery, where the subject of integration normally produces mega-decibel rhetoric from all corners.

Montgomery's normal static on education matters seems now to have migrated next door to Prince George's, where the teachers' union and the board are at each other's throats. The issue there is whether the teachers' new contract should include a binding arbitration clause, as suggested by a labor arbiter who intervened when negotiations between the board and union reached a stalemate over the summer.

Teachers had agreed to go along with other TRIM-strapped county employes and forego the salary increase, as long as they got the arbitration clause in return. The board is adamantly against binding arbitration and seems to be standing firm.

Chances of a teachers' strike are slim, according to observers. With 500 teachers laid off already because of shrinking resources, few union members seem anxious to risk their jobs by taking an action that is illegal.

But don't hold your breath for a solution to the contract. There will be a lot of sweat over this one, at least until the glacial winter settles in.