I think Americans can understand that judges draw on a variety of tools in interpreting the law, and that these tools differ for judges based on their constitutional values.
For example, when a statute is unclear, Justice Antonin Scalia might press harder on the language of the law, look at the context of specific words, and generally seek to understand what the written law means. He seeks to limit his own discretion, in part because the Constitution gives Congress, not the courts, the power to enact laws.
By contrast, Justice Stephen Breyer might focus on the purposes of the law and look to sources outside of the Constitution—including foreign law—to come to a decision. He may consider the outcome that makes the most sense to him because he considers judges to be a part of the democratic process. These are fundamentally different ways of dealing with difficult cases and they reflect two distinct attitudes about the proper role of a judge.
Ms. Kagan and those preparing her face a simple, political problem: "progressive" views of judging are difficult to defend. It may be why no recent nominee has tried. The simple statement that "judges should interpret the law, and not make it" resonates with Americans in a way that "judges should figure out the best answer" does not.
The reality may be more complicated than either of these formulas, but an attitude that emphasizes the rule of law has more appeal not merely because of its simplicity but because it captures the idea that judges are not policy makers. It emphasizes that judges should interpret the language of the law and try, as best they can, not to impose their own personal views of justice or the good when deciding cases.