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Early computer algebra systems, such as the ENIAC at the University of Pennsylvania, relied on human computers or programmers to reprogram it between calculations, manipulate its many physical modules (or panels), and feed its IBM card reader.^[1] Female mathematicians handled the majority of ENIAC programming human-guided computation: Jean Jennings, Marlyn Wescoff, Ruth Lichterman, Betty Snyder, Frances Bilas, and Kay McNulty led said efforts.^[2]

In 1960, John McCarthy explored an extension of primitive recursive functions for computing symbolic expressions through the Lisp programming language while at the Massachusetts Institute of Technology.^[3] Though his series on "Recursive functions of symbolic expressions and their computation by machine" remained incomplete,^[4] McCarthy and his contributions to artificial intelligence programming and computer algebra via Lisp helped establish Project MAC at the Massachusetts Institute of Technology and the organization that later became the Stanford AI Laboratory (SAIL) at Stanford University, whose competition facilitated significant development in computer algebra throughout the late 20th century.

Early efforts at symbolic computation, in the 1960s and 1970s, faced challenges surrounding the inefficiency of long-known algorithms when ported to computer algebra systems.^[5] Predecessors to Project MAC, such as ALTRAN, sought to overcame algorithmic limitations through advancements in hardware and interpreters, while later efforts turned towards software optimization.^[6]

A large part of the work of researchers in the field consisted of revisiting classical algebra to increase its effectiveness while developing efficient algorithms for use in computer algebra. An example of this type of work is the computation of polynomial greatest common divisors, a task required to simplify fractions and an essential component of computer algebra. Classical algorithms for this computation, such as Euclid's algorithm, provided inefficient over infinite fields; algorithms from linear algebra faced similar struggles.^[7] Thus, researchers turned to discovering methods of reducing polynomials (such as those over a ring of integers or a unique factorization domain) to a variant efficiently computable via a Euclidian algorithm.