The Montgomery County Board of Education voted yesterday to ask the County Council to increase spending on schools next year by $57 million, $1.6 million more than proposed by Superintendent Harry Pitt.

In all, the board added nearly 61 staff positions to Pitt's $576 million request for the school year starting this fall, for a total proposed budget of $577.9 million. To be added later are the anticipated costs of a salary increase being negotiated with school system administrators.

In his first spending request as superintendent, Pitt had asked for an increase of $55.5 million, or nearly 11 percent, over the 1986-87 budget. He had added few new improvements to the proposal, in an attempt to head off the kind of bitter confrontation over spending the school system had with the council last year.

The board also learned of the resignation because of poor health of first-term member Mary Margaret Slye, who has unable to attend many meetings recently. The board said it would select a member by March 16 to fill Slye's term, which expires in December.

The extra spending proposed by the board for next year would, among other things, further expand so-called Chapter I reading, language and math programs aimed at minority students; help reduce average class size by adding six elementary school teachers; pay for additional counselors and security assistants, and set up a centralized substitute teacher calling system.

The board also voted to increase Pitt's $96,500 annual salary to $102,750 and to add $200,000 for a pay equity study of the work performed by support staff such as food service workers.

In discussions in recent weeks about Pitt's budget request, several board members had called it too conservative.