showing that increased funding can provide better teachers, facilities and instrumentalities of learning (187 Misc 2d at 68). We agree that this showing, together with evidence that such improved inputs yield better student performance, constituted plaintiffs' prima facie case, which plaintiffs established. That the trial court's "Causation" section is largely devoted to the State's rebuttal arguments, rather than to plaintiffs' prima facie case, is insignificant, in that the court had already incorporated much of the correlation evidence in its discussion of inputs and outputs, as we have done. The trial court, for instance, concluded that teacher certification rates are one valid measure of teaching quality and are too low in New York City, and it founded these conclusions on evidence establishing the correlation between teacher certification and performance (187 Misc 2d at 26-27). The Appellate Division speculated about the significance of certification and noted that more certified teachers will be hired "in any event" (295 2 at 12), but its only clear holding about the quality of instruction -- which we reject -- was that principals' ratings of teachers should be the preeminent measure of pedagogical quality and, implicitly, that by this measure the teaching input is adequate ( id. at 13-14). The Appellate Division did nothing to undermine the rest of the trial court's syllogism: that better funded schools would hire and retain more certified teachers, and that students with such teachers would score better.[8] The same is true with respect to class size and instrumentalities of learning. We thus have no occasion to repeat the evidence establishing plaintiffs' prima facie case regarding the causal connection between better funding, improved inputs and better student results. The State nevertheless makes several further arguments concerning the correlation between its funding scheme and the educational results. Most of these points, however, more properly concern the apportionment of responsibility among various government actors than causation. In any event, the trial court interpreted CFE correctly when it said that the "law recognizes that there may be many 'causal links' to a single outcome, and there is no reason to think that the Court of Appeals 1995 opinion mandates a search for a single cause of the failure of New York City schools" (187 Misc 2d at 92). Socioeconomic Disadvantage. The State argues that poor student performance is caused by socioeconomic conditions independent of the quality of the schools and better remedied with investment in other resources. The Appellate Division agreed, reasoning that