I think you're reading the argument exactly backwards. The perception is that algebra (and advanced mathematics) is much harder than other subjects and the perception is that people can be good at a lot of things and still be bad at math. His argument is that in reality everything is taught poorly but it's easier to fudge the results when grading is necessarily subjective.
I am incredibly surprised that what I said was so controversial. Objective measures highlight (and therefore prevent or contain) fraud in all matters, and education is no different.
Yes, the argument that objective subjects are better judges of someone's education sounds plausible at first glance, but on closer inspection, I think the opposite is actually true. Suppose we have 10 completed math tests that have been graded, and all 10 are 100% correct. We should expect that the answers will be identical (or at least mathematically identical.) On the other hand, 10 essays that have all been graded as 100% would all be different.
One effect is that it is much easier to cheat at math than cheat at writing an essay. A good teacher can more easily identify if a student turned in work that they could never have written themselves. This is because each piece of writing is unique to the student. In math, a correct answer is a correct answer, so you can't distinguish cheating, which is why math tests aren't just about writing down the objective answers, but showing your work. Graders look at how you got to an answer, what techniques you used, whether you got to the answer in a straight-forward way or not. These factors are going to be much more varied between any two tests, and probably give you a better sense of who has mastered the material even when the final answers are identical.
To me, what this shows is that knowledge is highly personalized. Even when it is objective, and has clear right and wrong answers, the precise way you actually put knowledge to use is very much unique to each person. The fact is that understanding is a subjective experience -- everyone knows that there is a feeling associated with discovering the truth, figuring out a problem and so on.
"One effect is that it is much easier to cheat at math than cheat at writing an essay."
Interesting point, but the context is people failing algebra, and I don't immediately see how the ability to cheat contributes to that problem. Can you elaborate?
"Graders look at how you got to an answer, what techniques you used..."
And although free-form answers do make it a little "softer" of a subject than it may seem at first glance, there are good objective standards to go by. The fact is, a good multiple-choice test can tell you a lot about how well a student understands algebra; it's hard to devise such a test for subjects like writing.
The point about cheating is this: what makes cheating possible? It's because there is a gap between answers written on a piece of paper, and what is inside someone's head. This is true of all kinds of tests, but it is more true of answers that are identical for every student. If the names on the tests were somehow mixed, this would be undetectable by graders, where it would be immediately apparent for a written exam.
The significance of that has nothing to do with preventing cheating. It just means that the gap between right answers and real knowledge is greater when the answers are all identical. A concrete example: in my college physics classes, I studied with a friend who solved problems by memorizing which type of problem required which formulas. I figured out which formula to use by visualizing the problem, which is a much more efficient way of doing it and leads to a better general understanding of physics.
These important differences are undetectable just by looking at whether we both got the right answers on a test. You can account for this indirectly, by limiting the amount of time, so that my study partner would never be able to finish the whole test by using his method. But the information he gets about the incorrectness of his answers does nothing to help him fix his inefficient method. He did very well on the homework, the issue was only revealed at the midterm.
Physics education would be improved if taught people how to visualize problems, which means making it more qualitative and less quantitative.
The issue is teachers cheating by grading more easily than they should, not students cheating. You are completely missing the point.
Graders look at how you got to an answer, what techniques you used, whether you got to the answer in a straight-forward way or not. These factors are going to be much more varied between any two tests, and probably give you a better sense of who has mastered the material even when the final answers are identical.
For the most part, they don't. I taught calculus for several years, and different students picked different techniques. I observed no correlation between particular choices and student quality. Good students answered more and harder problems, bad students answered only a few easy ones.
The only reason we demanded they show work is to make sure they solved the problem rather than copying the answer of the guy sitting next to them.
