The infantile project began to slowly grow. In the 1984-1985 school year, Moses added nine eighth graders and seven seventh graders to his class. This was the school year in which some of the basic concepts of organizing came into play. Parents became involved with the Algebra Project and supported it with their time and effort.

The spring of 1985 was a key moment in the history of the Algebra Project. The Cambridge School Committee officially recognized the project as a component of their school system.

Another milestone was reached in the spring of 1986 when the Project's first full class graduated from middle school and 39 percent of the class went on to honors geometry or algebra in high school and no one in the rest of the class took a reach back math course.

Moses had studied the Philosophy of Mathematics at Harvard University. One of his teachers was Professor Willard Van Orman Quine. As Moses' refined his curriculum for the Algebra Project, he drew on some of Quine's ideas. One of these ideas was that people need to first understand math through everyday language and then they can translate that into the more abstract language of mathematics.

One of his first implementations of this idea was his use of the Red Line, the rapid transit line, which carries passengers between downtown Boston and Harvard Square in Cambridge. In a classroom exercise the subway line represented a mathematical concept, a number line.

In 1989, Moses solicited grants to expand the Algebra Project outside the Cambridge School system. At the same time teachers and administrators in the Boston Public School system were contacted. Three Boston school expressed an interest in the Algebra Project. The result of their interest was the formation of the Algebra in Middle Schools (AIMS) project. In October of that year, AIMS received a small grant from the Boston Foundation to fund this project The Boston Public School administration provided support in the form of hiring substitutes to fill in for teachers who were being trained in the Algebra Project Curriculum. In three months the Algebra Project was able to train fourteen teachers and one administrator in its curriculum for AIMS.