For years, the debate over public education reform has centered on fin

Very interesting discussion going on here. Let me add my two cents to it. Let's first dissect the passage, line by line:
For years, the debate over public education reform has centered on financing - It's a background statement for the passage. It gives a background that a debate has centered on financing. Then, the passage given two counter-views of the debate.

Many claim that pouring more money into the public schools will improve student performance. - This is view of one side of the debate.

However, the only way to fix our school systems is to inject new ideas and new approaches. - This is view of other side of the debate. The author is on this side of the debate. The use of word "only" makes this view completely counter to the other view. Presence of "only" means that this view of the debate means two things: new ideas and new approaches are needed to fix the system and pouring of money "cannot" fix the school systems.

Today the schools are organized to benefit their adult employees rather than the students. - This opinion is used to mean that the current school systems are not doing what they are supposed to be doing and thus, they need a fix. In addition, "... benefit their adult employees rather..." could also mean that the way (financial) resources are used currently is benefiting employees rather than the students. Thus, this opinion would also support the viewpoint that putting more resources will not help; new ideas and approaches are required.

The question asks us to find an option which most weakens the argument. But what is the argument here?
From our above understanding, the argument is like:
Conclusion (Claim): new ideas and new approaches are needed to fix the system and pouring of money "cannot" fix the school systems.
Premise (supporting opinion): the way (financial) resources are used currently is benefiting employees rather than the students

Now, let's look at each of the options:
A. Schools that have instituted “new approaches” attract the best performing students. - Supports the conclusion rather than weakening it.
B. Schools without outside playgrounds have lower levels of student performance than schools that do. - It means that students with outside playgrounds have higher level of student performance than schools that do not. So, if the absence of outside playgrounds in schools is due to lack of funds, then putting more financial resources will help them to build playgrounds and thus, achieve higher level of performance. So, by making an assumption (that the absence of outside playgrounds in schools is due to lack of funds), this statement works as a weakener.
C. Studies show that student performance corresponded most directly with the education of the students’ families. - Generally saying, education of students's families is not being talked in the passage. Even if this statement is considered correct, it could mean two things in different scenarios:
1. Scenario One: If education of families can't be changed - In this case, we can't really do anything to improve student performance, which makes the whole debate irrelevant.
2. Scenario Two: If education of families can be changed - In this case, we can work on educating the families while simultaneously working with students. Doing such a thing could be characterized as a "new idea", which would support the given argument.
Thus, this statement doesn't weaken the argument.
D. School employees, by an overwhelming margin, said that the system performed well. - This is irrelevant. Opinion of school employees on the argument is irrelevant.
E. Researchers in education have shown that students from school districts with high per-capita spending tend to receive higher scores on standardized tests. - If higher capita spending means higher spending per student and if higher scores on standardized tests means higher performance, then this statement weakens our conclusion that pouring of money cannot fix the system. So, by making these assumptions, this statement can act as a weakener.

From the analysis above, we see that both options B and E can be weakeners, if we make the required assumptions. However, we need to select the options which weakens the argument the most. Therefore, we need to find the stronger weakener of these two.

A strong weakener is one which weakens the argument without making any assumptions. In this case, both the statement make assumptions. The stronger of the two would be the one whose assumptions are easy to justify within the context of the argument.

In this case, I think assumptions for option E are easier to justify:
- Assumption 1: Since we are primarily talking about public finance, per capita spending should refer to spending per student by the public machinery, rather than spending per individual
- Assumption 2: Generally whenever performances are measured and compared, they are through standardized tests. Thus, higher scores on standardized tests should mean higher performance, in this context.

In case of option B, absence of outside playgrounds could be due to reasons other than financial ones. It's actually not easy to justify that the only reason for absence of playgrounds would be lack of financial resources.

Thus, the correct option should be E.

Cheers,
CJ