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I'm preparing for a university entrance exam that has different subjects (like Biology, Mathematics, Physics, Chemistry ,...). In the Math section, there are $30$ problems for $47$ minutes ($94$ secs per problem in average). I'm wondering when I'm solving math problems in home preparation, how much time should I spend on each problem? On one hand I've heard a lot that the more time we spend on a problem we can benefit more but on the other hand, because it is a timed exam and I can't spend much time thinking on a problem during the exam, it may not be appropriate to deep dive into thinking in preparation too. So may you please help me for this situation?

Edit:

It is a multiple choices exam in high school level but problems got more challenging recently. I've asked some of the problems on the site before. Like here , here , here, here,here and here.

Amirali
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  • I’d say practice slowly and patiently, until you understand the material so clearly that the problems become easy. – littleO Apr 29 '22 at 21:11
  • Would need more information like what level of maths you are and what level of maths the test it aimed towards, and what type of test it is (multiple choice?) to give a serious answer. But in general I think one should try to improve their understanding to the maximum and then improve speed after by timing yourself and writing the same answer over and over again, reducing the amount of time you allow yourself to write the answer, for example. – Adam Rubinson Apr 29 '22 at 21:11
  • @AdamRubinson I just added more context in the question body ;) – Amirali Apr 29 '22 at 21:31

2 Answers2

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I disagree with the other responses, and regard the situation as the reverse of chess preparation (at least as I understand it). My understanding is that excelling at classical chess time controls will improve your ability at blitz chess, but not vice versa.

For me, the key to maximizing your grade on a timed multiple choice Math test is that you don't have to know the answer to a problem in order to answer the problem correctly. I am assuming that your Math knowledge/learning is proceeding at a normal pace, so the real issue is how to over-achieve your test grade, with respect to your Math knowledge.

I refer to this as Meta-Cheating. Although the term is not generally used, the pertinent skills are very common at the university level, in multiple choice tests on any subject. To a certain extent, using the practice tests analogizes to practicing how to play a musical instrument. You should develop the right feel to using your time efficiently.

I advise using the practice tests, as follows:

  • You definitely want to be wearing a watch, preferably one that has a stop-watch mode.

  • You want to devote $(1/2)$ of the allotted time for your first run through of all of the questions. For example, if you have $47$ minutes to do $30$ problems, you should pace yourself to complete the first run through in $23$ minutes.

  • During the first run through, for those questions where you are sure, mark your choice, and mark in the margin a check mark. Alternative, note the question number on a separate piece of scratch paper, with a check mark.

    For those questions that you suspect that you know the answer, mark the question with a ?, and go ahead and guess the answer, marking your choice.

    For those questions where you are more or less lost, leave the choices blank, and mark the question with a $(0)$.

  • At this point, you should have about $(24)$ minutes remaining. Further, at this point, your subconscious should be sending you messages about the intent of the problem composer. Presumably, the problem composer will intend that each question has an educational value and that wrong choices will represent an improper use of the appropriate formula.

  • At this point, with $(24)$ minutes remaining, you should probably receive some messages from your subconscious that are repetitive. That is, where the educational value of (for example) question-1 is the same as the educational value of question-10.

    This means that if you know the answer to question-1, and don't know the answer to question-10, you can use your knowledge of the answer to question-1 to make a high probability guess as to the answer of question-10.

  • You should generally assume that problem composers lack the skills of either professional poker players, or high-level game theorists. This means that the problem composers won't generally be trying to trap/penalize meta-cheating.

  • When you go through the exam the 2nd time, the elapsed time should proceed from about the $(23)$ minute mark to the $(35)$ minute mark, which is why your watch is important.

    During the 2nd run through, I advise spending about $10-15$ seconds on each of the problems that you have marked with a check mark.

    This will serve $2$ purposes. It will allow you to catch the occassional arithmetic mistake or problem-misinterpretation mistake. It will also reinforce any messages coming to your subconscious re the educational purpose of the questions that you know the answers to.

  • For the rest of the questions, you want to spend about twice as long (perhaps 40-60 seconds) on the ones that you marked with a ?, as you do on the ones that you marked with a $0$. For each question, at a minimum, try to eliminate any of the choices that you know are wrong, and then make a reasonable guess as to the right answer.

  • At this point, it is assumed that you have finished the 2nd run through, and have about $12$ minutes remaining. Then, the remaining portion of your time should be spent quickly reviewing all questions, hoping to isolate specific questions where you (for example) are now able to eliminate a choice that you know is wrong. This should allow you to increase the probability of answering the question correctly.

When it comes to arithmetic intense problems, note that problem composers are usually biased towards arranging the parameters so that the numbers in the solution look nice. So, if your computation results in $\sqrt{101}$, which does not match any of the choices, then as you are running out of time, $\sqrt{100}$ may represent a high probability guess.


This is the point in my response that I risk violating my religious principles, re getting an A on a test when you know nothing. It is probably a good idea to begin the metaphoric violin practice $1$ month in advance.

If possible, tell the teacher that you would like to review practice exams from the last $(10)$ years, that contain problems that are similar to but different from the problems that will be on the official test.

Hopefully, you will be able to find $5$ such practice exams. Take such an exam, and then set it aside for $(24)$ hours. Then, research each question, with special emphasis on questions that you did not originally know how to answer. Spend a couple of days focusing on the answers. If necessary, try to derive the answers yourself.

Then, re-take the exact same test, but only allowing yourself $(24)$ minutes for the entire test. This test is on questions that you have just researched. Work through each of the problems, from scratch, quickly. This speed drill should also improve your learning in the particular area where you were deficient.

Then, move on to the 2nd test, then the 3rd, and so on. With each test, go through all of the steps, as before. While it is understood that your goal is to improve your test-taking skills, you should relax any private religious principles that you might have against actually learning the material.

user2661923
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  • Wow Thanks a lot for spending your time to write such a detailed answer! I've read the first paragraph so far: I'm a chess player too;) sometimes condition of the exam make it like a bullet game! – Amirali Apr 29 '22 at 23:17
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I suggest that the best way to study for this kind of math exam given that you have sample problems like those that will appear is to start each problem with the goal of solving it quickly. If you succeed, move on to the next. If you don't, then spend the time to do a deep dive. After you've solved it with effort, look back to see how once you know the answer you could have seen it more quickly.

As you move through the problems you might see patterns in the way they are constructed that hint at patterns for quick solutions.

If the exam is multiple choice (common for timed exams with many short questions) you can learn ways that eliminate some answers quickly. Checking each plausible answer might be faster than solving the problem and seeing which answer matches what you found.

Note: this is specific advice for this specific situation. It's not advice on how to learn mathematics.

Ethan Bolker
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