The American constitution is notoriously difficult to amend. If you want the cold, hard, figures, the sobering statistic for a prospective constitutional amendment is less than a .25% success rate: 27 successes in over 10,000 attempts. This figure is far beneath the amendment rates for other countries around the world. And the result is that many amendments that command majority, or even super-majority, support, do not end up clearing all the hurdles of Article V (think of the ERA, flag burning, or school prayer amendments).
The result is a constitution that runs a perpetual risk of being out of sync with majority perspectives. To resolve this, while still maintaining the countermajoritarian ethos that motivated the difficult amendment process to begin with, Professor Rosalind Dixon proposed the idea of "partial constitutional amendments", which would be given persuasive but not controlling authority by the courts. This proposal, though novel, has some parallels in the constitutional jurisprudence of other countries (such as Canada and India), whose courts tend to see themselves in more of dialogic relationship with their respective legislatures.
A partial constitutional amendment is simply an amendment that achieves majority support, but for one reason or another does not clear all of the requirements of Article V. It might not achieve a supermajority, or it might not achieve ratification in a supermajority of states. These amendments should be seen as akin to persuasive authority -- the further along they are in the process, the more persuasive they would be. The idea is to provide a mechanism for constitutional updating that is respectful of the Article V process, has a clear democratic hook beyond personal judicial preference, and allows for more rapid synchronization between democratic preferences and constitutional doctrine.
Professor Dixon is operating within a tradition of thought holding that courts generally are relatively responsive to sustained democratic preferences, at least eventually. However, they also labor under a desire to not be seen as making a decision "under fire", that is, solely writing a decision to placate democratic outrage at a prior (presumably now-overturned) precedent. The result is that constitutional updating occurs fitfully and perpetually under a shroud of potential illegitimacy.
Consider the reaction to and eventual reversal of Hammer v. Dagenhart, which struck down a law forbidding the transport through interstate commerce of goods made with child labor. Decided in 1918, the decision met with immediate outrage and an attempt at an Article V override. The effort eventually failed, but it did get through Congress, and was ratified by 20 states within a decade of its proposal. Meanwhile, Congress issued several new laws targeting child labor, which were successively struck down by a recalcitrant court in 1919 and 1923. The court managed to maintain its resistance for 20 years beyond Hammer's decision date. The first chink in the armor came with West Coast Parrish in 1937, but Hammer wasn't formally overruled until 1941.
Adopting a policy of partial constitutional amendments could potentially shave valuable time off this process. By providing a formal avenue by which congressional responses to unpopular judicial decisions could be given weight and authority by the court, the partial constitutional amendment would help dissipate some of the perceived legitimacy concerns where courts seem like they might be too beholden to the democratic branches. In a sense, it formalizes a modernizing process in the judiciary that many scholars believe exists, but few judges are comfortable admitting to.