Normative science is a type of inquiry by which people seek to determine which goals are good, and which means toward those goals are good. Logic, Mathematics, Philosophy of science, and Ethics are all normative sciences, because they seek to provide norms and standards by which phenomena can be interpreted and effective action can be taken.
Normative sciences are extremely important, because our day to day actions are only as good as the normative values and methods that guide us. Without the study of justice, there can be no law. Without philosophy of science, there can be no science. Without logic, there can be no learning.
These rules of logic, mathematics, and philosophy of science provide such guidelines to the physical sciences, and therefore are the foundation of all sciences. Scientists apply hundreds of logical, mathematical, and philosophical rules in exploring the natural universe. If they apply the wrong rules, they will get the wrong results.
The difficulty we run into is that the normative sciences cannot be proven as we commonly understand the term. One cannot "prove" that logic is true without using logic to prove it. Logic is a matter of Faith. Only when one takes the leap of faith to believe in logic and apply logic does one find it to be true; and by experience, one's logic becomes stronger, and better able to understand the cosmos.
Sadly, the normative sciences are badly neglected in modern western civilization, in part because they are not as "profitable" as the other sciences which can lead to the invention of cars or cell phones. The dangers of technology without normative sciences were seen throughout the 20th century, as enormous strides were made in technology, and that technology was in turn used to slaughter hundreds of millions of people, drive thousands of species to extinction, and do a great deal of damage to the Earth. Failures in the study and application of ethics have led to skyrocketing crime and drug abuse and the decay of the family. As a result of widespread flawed logic, many people are unable to grasp the incredibly powerful evidence for creation, and choose instead to believe a theory of evolution so fundamentally flawed and illogical as to be laughable if it weren't so tragic.