How to analyze the level of difficulty of mathematical computation of a problem on a standard mathematical test designed for high school students? I mean how to choose some indices that can reflect the difficulties of mathematical computation without giving a test in advance?
The difficulty of a problem depends on a large number of factors. Different students are going to find different problems difficult (for example, some students struggle with computation but do well on conceptual questions), and the material that you actually cover in lecture and homework is going to influence how easy or hard some problems are (to give a higher level example, half-angle formulae for the sine and cosine functions are relatively easy to prove if you already know Euler's formula; they are a little harder to prove if you can only use properties of the unit circle).
That being said, there are a number of things that you can do in order to get a sense of how difficult an exam problem might be:
- Use homework exercises from a text as a model for exam questions (see this answer). In addition to being organized from easiest to hardest (typically), these problems have probably been vetted by more than one person, edited to death, and checked carefully for alignment with the text. This means that the level of difficulty should be fairly easy to assess (as per guest's answer) and that problems from the text (assuming that you are using that text) should align well with the material being taught and be "fair" questions to ask.
- Work the problems yourself (see this answer). This can't really be used to check if problems are "too easy" (since you might have inadvertently created problems that don't align well with what you taught), but it is certainly a good way to check if your problems are too hard.
- Exploit your colleagues. Unless you are in a very small school out in the middle of rural America, I would guess that you have colleagues that you can talk to (you say that you are designing a test for high school students, so talk to the other people in your math department). Have them look over your exams and give an opinion about the difficulty. If they have a lot more experience than you, they will likely be able to guide you quite a bit. Even if they don't have that much more experience, they can still point out flaws or problems that are potentially too difficult.
- Consider student performance. This isn't really an a priori measure of difficulty, but it can be used to design better exams in the future. Before you give an exam, determine how hard you think each question is. After the exam, look at student performance on each question. Are there questions that nearly every student got right or wrong? Are there questions where students clearly burned through a lot of scratch paper or made common errors? By understanding where students struggle on one exam, you gain information about how to write future exams.
- Finally, as hinted at by the previous two answers, let experience guide you. If you are a young or new teacher, you are going to write some terrible exams---most of us do that. Learn from your mistakes, consult your colleagues, and you'll do better the next time. In particular, as you give more exams, you will have a better idea of which problems are easy or hard for your students.
Answer by guest in a comment:
Give problems modeled on the homework drill problems. Typically they will be divided into 2-3 sets of increasing difficulty by number. (a) simple plug and chug (b) longer multistep problems (c) challenging problems requiring combining techniques, proofs, or deriving something new not taught yet. Tests should avoid much or any (c) problems.
An answer originally by Dan Fox in a comment:
One way to judge computational difficulty is by performing the computations oneself. With exams at the university level, my rule of thumb is that if I set a two hour exam, I should be able to physically write completely correct solutions in no more than 15-20 minutes (otherwise the exam is too hard). At lower levels, I would suppose the teacher should be able to complete (doing and writing all the required computations) whatever is tested in an hour test in a few minutes.