O.K., let's check some numbers.
The report from Hanushek, Peterson, and Woessmann (in PDF!) offers a nice table giving percentages of kids getting advanced math scores in each state. The Census Bureau offers data on per-pupil education spending in each state. I paste those numbers into Excel and run the CORREL function. The results:
|Correlations with percentage of students scoring "advanced" in math|
|Salaries and Wages||0.40|
|Instruction Employee Benefits||0.33|
|Support Services Total||0.33|
A correlation of 1 means an absolute postive relationship (when one number increases, the other one always increases). A correlation of 0 means absolutely no relationship, or complete randomness (the numbers have no more connection than a list of results from rolling dice).
A correlation of 0.37 means there's some positive relationship, but that other factors can wash out that relationship. That fits what we see when we line up the states by per-pupil expenditures:
|State||per-pupil expenditures (2008)||% adv. math scores|
Among the top ten big spenders, you have six states above average on the advanced math score and four below. Among the bottom ten low spenders (that includes South Dakota!), you have three states above average and seven below. That's far from a perfect correlation, but it does appear that, all other things being equal, if you make more money available to schools, they'll have a better shot at turning out smarter kids.
Now look again at the first chart, which breaks down the correlation by how the money is spent. The strongest correlation in the batch is the 0.42 you see between advanced math scores and teacher pay. The weakest correlations are the 0.17 for pupil support and 0.18 for general administration. Again, none of these correlations are guarantees of results, but if you have money to bet on improving student performance, these numbers suggest the best place to put that money is right into your teachers' pockets.
Of course, the problem is that to understand this argument, we need more kids who get good math scores....