The concept of egalitarianism is much bigger than education theory. It has its origins in enlightenment thinking, and is entrenched in the U.S. Constitution, in the phrase "all men are created equal." The earliest notions of equality, those of 17th and 18th century classical liberalism, were more consistent with the notion of same opportunity than same outcome. Definitions of equality based on the positive right to certain material resources became more popular in the 20th century, with thinkers like John Rawls being champions of these definitions.
"Equality of opportunity" is the oldest and most widely accepted definition of egalitarianism. Under this conception of egalitarianism, all people have the right to participate in -- or at least to be considered for participation in -- the same institutions. In the classroom, equality of opportunity means giving all students access to the same basic educational opportunities. Students should not be blocked from participation in school activities due to gender, race, ethnicity, sexual orientation or socioeconomic status. In an egalitarian classroom, the only factors that should limit a student's opportunities are merit and academic achievement.
A newer conception of egalitarianism states that people are equal when they achieve the same outcomes. This conception of egalitarianism is somewhat more controversial than the opportunity definition, since it can be seen as undermining meritocracy. In an egalitarian classroom based on equality of outcome, marks may be abandoned in favor of constructive criticism and guidance, and students who struggle may receive extra attention. In cases where grades are used, they do not block students' abilities to participate in class activities.
The capability approach to egalitarianism accepts equality as an ideal and measures it against an individual's ability to contribute. In an egalitarian classroom based on the capability approach, all students with the same capabilities have access to the same educational opportunities, but efforts are not made to accommodate students in activities outside of their proven range of competence. The capability approach is related to equality of opportunity, in the sense that both approaches accept that certain things must be earned. However, the capability approach is much stricter about this: it does not accept that all students should be given the same opportunities, unless all of the students have the same abilities.