But in this case it's not the rational thing to say when you use decision theory.

I disagree. The proper response to not knowing the answer is to admit to not knowing and then give your best guess, not to try to hide your ignorance, because if you succeed then the teacher doesn't know you need help. A student who is more concerned with not displaying ignorance than with not being ignorant is not trying to learn, which is not rational. That which can be destroyed by the truth should be, and it probably won't be if you try to avoid finding out what the truth is.

You can easily do worse than maximum entropy if you guess at random.

If you base your guess correctly on an incomplete model of reality, which you've constructed correctly from past observations, you can never do worse, on average, than maximum entropy. More evidence can never lead to less information (as per the Data Processing Inequality).

The key phrase here is "on average". If you guess at random from all possible explanations of a given phenomenon, you will, on average, die before guessing the correct answer. There is a reason the monkeys with typewriters are given infinite time to reproduce Hamlet.

Moreover, as the set of answers considered increases in size, the expected utility from giving any one answer tends towards the expected utility of a wrong answer. As long as giving the wrong answer gives less utility than admitting ignorance, admitting ignorance is almost always the utility maximising option if you don't know.

Furthermore, "getting it right" [...] does not necessarily mean that you possess any anticipation-controllers.

On the contrary, it mean exactly that. Being rewarded for predictive powers improves your model of the world, whereas "I don't know" is an excuse for not knowing.

If I write down a number and then take a number from a table of random numbers, and the numbers are the same, does this mean that I'm psychic? Because if getting the correct answer means that I have useful anticipation controllers then I must be.

"I don't know" is not an excuse for not knowing. That makes no sense whatsoever. "I don't know" is a statement about whether I know something or not, not a statement about whether I ought to know. If you can't admit fallibility then you will never learn anything.

The points you make about the benefits of testing students' knowledge are true. Unfortunately, they miss the point - while it is important not to penalise guessing incorrectly, so as not to dissuade admitting ignorance, it is much better to actively reward admitting that you have tried and failed. If a confused student does not always seek an explanation, the reward for seeking explanations isn't large enough yet. If students are content to remain ignorant, something is seriously wrong with your system for making students less ignorant.